You need to establish a pecking order, one where you are at the top. Your cats need to abide by your rules, and that includes not picking on each other. Even if they cannot stand each other, they need to respect your wishes and not start fights.
1. No picking on each other.
You need to consistently enforce this rule in all directions. Cats don't respond to your intention of teaching them (dogs do), but cats do respond to their environment and situation.
A cat might never stop doing something because you tell it to, but you can get it to stop doing something because it causes you to retaliate
Disclaimer:
A retaliation by you is supposed to be negative feedback and an inconvenience to the cat. It should never cross ethical boundaries. No violence, fear, or harsh punishments.
The rule is clear, no picking on each other. If you can't interact nicely, then don't interact. All you need to do now is consistently enforce it.
2. Responding to transgressions.
A real world example might help here. We have three cats: Cleo, Misha and Alfie. Cleo and Misha are sisters (about 15 months old), Alfie is a brand new addition (about 6 months old). Alfie always wants to be first, everywhere, every time.
Treats? Alfie is front and center. Food? Alfie rushes to the first bowl and quickly jumps to the next bowl when his is finished. I'm giving love to a cat? Alfie will come and ask for attention.
I may be biased because I like the little bastard, but imo Alfie does not mean to undercut the girls, he's just eager for everything. He is always highly suggestible and easily distracted by anything that catches his attention. Literally anything distracts him, both in cases where it is a benefit or a drawback. He's not gaming the system, he's just incessantly eager.
However, and this is key to how to respond to transgressions, Alfie's intent does not matter. He gets punished for breaking the rules, not for why he broke the rules. The rules are non-negotiable, and need to be enforced as such.
He's allowed to be highly energetic and eager. Cleo and Misha are a bit more relaxed, so it's natural for Alfie to generally be first. I don't punish him for being first if the others are slower than him. However, whenever he crosses the line and actually cuts in between me and another cat, he does get punished.
I (softly) put my hand on his side and then give him a strong push. At the very least, he'll slide to the side. At worst, he'll fall on his side. It doesn't hurt him, but it inconveniences him to be pushed away and have to walk towards me again.
If he rushes at a treat that was clearly placed in front of another cat (I tend to also say the name of the cat whose treat it is, for further clarity) I will do everything in my power to ensure that the treat is eaten by the cat I gave it to. I will take it from Alfie's mouth if I have to (luckily, he doesn't put up a fight).
The core of my feedback is always consistent: No cat gets to bend a rule that I have made. If I give a treat to Misha, no one gets to take it from her.
This also extends to unjust behavior that I see happen. If a cat drives another cat from their sleeping spot in order to claim it for themselves, I will actively remove that cat and make sure their crime did not pay. This includes dunking water on the spot (after the cat has been removed) to make it an inhospitable sleeping spot.
Interestingly, before Alfie joined our family, Cleo was similarly eager. It was always "me first!" with her, and we needed to push her aside to actually be able to interact with Misha. However, Alfie is much more energetic (and socially oblivious) than Cleo, and now Cleo is basically being treated the way she (obliviously) treated Misha. Since Alfie, Cleo no longer cuts in front of Misha, not even unintentionally.
This sort of proves my point: cats learn to not undercut others (unintentionally or intentionally), when they themselves become the victim of being undercut. When I see a cat undercut another cat, I will interfere in a way that the offender becomes a victim to their own crime.
3. Patience and consistency.
Most people who see me interact with my cats make a comment about how much effort it must take to enforce rules that get broken constantly, since cats are not really trainable (compared to dogs).
But I disagree with that. Cats are slow learners (compared to dogs) because they do not observe a human's intent to teach, but that doesn't mean that cats are unable to learn something.
Consistency is the most important factor here. If you always apply the same rule, and apply the same punishment, then the cat will eventually see the pattern and learn to avoid it.
Cats already learn this way. There's a rope hanging off of a bucket. The bucket is standing on a table, filled with water. Cleo jumps and tries to catch the rope. The inevitable happens: she catches the rope and throws a bucket of water on herself. Cleo made the mistake a second time, but has not played with the bucket since then. Cause meets inevitable effect. Cleo saw the pattern (play with the bucket => get drenched) and actively avoided it in the future (don't want to get drenched => should not play with bucket).
The key part here is inevitable. It's not just a random occurrence, the effect consistently happens. I try to artificially recreate that same environment of cause and effect. Alfie takes food from another cat, and gets pushed aside (roughly but without hurting him). The more aggressively he tries it, the more aggressively he gets pushed back.
Misha sometimes walks away from treats. When she does, he's allowed to take it once she turns her back, unless he has already tried to steal it.
It has taken about 2-3 weeks, but Alfie understands that now. He will eagerly look at the treat and Misha, but he will generally not try to steal it anymore (exceptions made on days where he's much too playful to be thinking straight, he's still a kitten after all).
4. Focusing on your situation.
My situation has been a bit more relaxed. It's mostly about a social fax-pas. Your situation is more severe, your cats are antagonizing each other. You can apply the same principles as above, but you'll have to tailor it to the situation.
The rule: No cat shall be aggressive or unkind towards another cat. No cat is above this rule. As the pets' owner, you will not accept any disregard for this rule (no matter which cat is breaking the rule).
Kitty's favorite activity is chasing Faith and battling with her. ... Faith is weak and old so she's the perfect target for our little bully.
The moment he acts out, you respond. This can be done by name calling (neutral, then angrily), throwing soft things (e.g. pillows) or physically intervening. The important part is that Kitty is guilty, and must be appropriately punished. I'll address possible punishments further in the answer, but the general idea here is that you need to tailor a punishment that fits Kitty's crime. Crime does not pay, and it is your job to deliver that message to Kitty's doorstep.
He's gotten into quite a few 'duels' with Pip (all of which Pip won) and so i think he's trying to show his dominance. He failed with Pip.
If you see it happen, you need to intervene. Regardless of who started it, if both are actively engaging each other in the fight, they are both guilty. Note that you should observe what is going on. If one of them is attacking, and the other is fiercely defending (but doing nothing more than defending incoming attacks), then only the aggressor is guilty. But if both of them are aggressive (even if one of them started off simply defending themselves), both are actively breaking your rule and need to be punished for it.
If Kitty is the aggressor, and Pip consistently wins before you can intervene, you do not need to intervene after the fact (as long as no cat gets hurt in the process). Kitty has already experienced the effect from what he caused: Pip dominated him. Kitty must recognize the pattern and decide: do I want to keep losing this? Or do I want to avoid poking the bear?
However, if the aggressor ends up winning, then you do need to intervene after the fact and make sure that the aggressor does not gain anything from their behavior.
I've not been sure on if i should ask this or not but the final straw was when i heard yowling and hissing from outside today.
In order to retain consistency, you should yell at the hissing cat before you even know who it is. That proves the point that you're not upset with a specific cat, but rather whoever is being unkind.
whenever Faith comes in Kitty tries to chase her so we have to hold him down so he won't chase her.
Holding him down is a good idea. However, maybe a pedantic niggle, but I'd make sure to only hold him down after he has shown to make a move. If you immediately hold him down before he has even misbehaved, then there's no lesson for him to learn. However, if he only gets held down after misbehaving, then he can learn the pattern (run after Faith => get held) and learn to avoid it (don't want to get held => shouldn't run after Faith)
Don't be afraid to hold a pillow and use it as a projectile to block him when he makes a move for her. If he's being that aggressive that you can't easily stop him, then you're morally allowed to respond in equal measure by throwing a pillow (I would aim it in front of him rather than on him).
Even if the pillow doesn't stop him, it still serves a purpose as a signal that you are telling him off. It also has the added benefit that you holding a pillow will eventually become a reminder of what will happen if he misbehaves.
Things get worse when Faith fuels the fire. Whenever she sees Kitty she hisses and tries to scratch at him.
Faith's reaction is understandable. But just like how I said that Alfie's lack of ill intent is irrelevant; so is Faith's justification for hissing. Hissing is unkind, and unkindness does not get tolerated.
You may think that this is unfair on Faith. But keep in mind that you're also teaching Kitty the same lesson at the same time. He sees that other cats get punished for the same misbehavior. This causes him to realize that getting told off isn't something that's unique to Kitty himself.
Again, you are showcasing a pattern (cat acts unkindly => cat gets told off) that the cats will eventually learn to avoid (cat doesn't want to get told off => cat shouldn't act unkindly).
Things get even worse when Pip joins in though. Pip doesn't know what he's doing, he chases her very rarely and i think he only does it because Kitty does it. (Kitty is then ultimate bad role model.)
I think the pattern is starting to be clear. Pip is guilty. Pip gets punished. Kitty is not to blame for being a bad role model. Pip is to blame for following Kitty's example, especially if he has also seen Kitty getting told off for this misbehavior (and should therefore know better).
I've tried everything, clapping, flicking Kitty/Faith (when she starts the fight) on the forehead/nose, nothing works.
I think I'm coming across as a general hard-ass. Keeping that in mind, I'm not a fan of physical punishment such as flicking on the nose. As I see it, only a human is able to flick a cat on the nose, and thus it leaves the possibility for the cat to blame humans for being flicked on the nose.
I push Alfie, because he pushes the other cats. When he bites them (more than just horseplay), I slightly pinch him when I grab him. When he chases them out of their comfy spot, I do the same to him. Although I'm fighting fire with fire, I make sure to always respond in equal measure and not escalate needlessly. I would rather inefficiently push Alfie instead of "play the human card" that will always trump Alfie.
If you respond too strongly, the cat will see you as the aggressor. Instead of learning a lesson, it'll learn to avoid or dislike you. If you never go beyond the cat's own transgressions, they can't blame you for something that they just did themselves.
What I expect from your cats, long-term.
This is just inference from what you've told us. Take this with a grain of salt, I don't know your cats other than what you've told me (and even then I could've misunderstood.
Faith seems to want to be by herself. Whenever Pip or Kitty misbehave, I expect that you'll end up enforcing a peaceful atmosphere for Faith (e.g. by separating the offender temporarily).
Pip seems to have no issue responding to Kitty in equal measure, so you need to defend him less. But when he misbehaves like Kitty, you should punish him like Kitty. He needs to see that the punishment is directed at the behavior and not just at Kitty.
Kitty is going to need the most work. He is going to have to learn how to back off. As a general rule, you can never stop a cat's misbehavior, but you can steer it. You can't stop a cat from scratching things, but you can redirect them to a scratching pole when they target your furniture or couch. The same is true for any other instinctive behavior.
If Kitty's misbehavior comes from playfulness (which Faith does not reciprocate), try playing with Kitty to redirect his energy. Even if he's not trying to play with Faith (but likes playing with you), you can play with him to tire him out and make him less likely to chase Faith.
If Kitty is being aggressive for the sake of aggression, then that needs to be adamantly addressed. Do not put up with that behavior, even in small bursts. Do not allow him to be unkind to the other cats in any way, shape or form. Make sure that doing so leads to an inevitable punishment that he will regret (but don't overreact either).
5. Fair punishments.
In order to not lengthen an already lengthy answer, I'll simply link to an earlier answer I gave on this topic. Chapter 4 specifically addresses the escalation pattern that I use to teach my cats how to steer their own behavior (by making them want to avoid you steering their behavior for them).
