A trick (that worked in the past for me) is to take a long walk with the 2 problematic dogs. Each of you should take one dog and then you just start walking with the both of you in the middle, separating the dogs. They'll probably want to interact with each other right away, but at this stage you keep them strictly separated and just walk on. You shouldn't speak too much with the dogs or give them commands or anything. Hold the leash short enough that the dog cannot walk around you, then nudge him away from you every time he wants to go the other dog.
It'll take a few minutes for both of them to calm down a little and concentrate on the actual walk rather than each other.
When they have calmed down a little bit, you allow then to walk next to each other and interact if they want. But don't encourage them or anything, the dogs must approach each other on their own volition. Your job is to walk on.
The psychology behind that is to indirectly tell the dogs that they are a pack now. Packs move together in the same direction and are friendly towards each other. Both dogs already accept you as part of their pack. Now you introduce an additional dog to the pack and make them act like a pack, even if they wouldn't do that otherwise.
I once saw this method in "Dog Whisperer with Cesar Millan" and tried it with my own dog. At that time she wasn't socialized with dogs and wouldn't allow any dog near her, but after such a walk, she was very relaxed around the dog of my friend.
Another very big advice is to learn dog body language. That will give you the knowledge to assess whether or not your dogs tolerate each other or when situations might escalate into a fight.
I've seen a puppy and old dog interact once: the puppy wanted to play and the adult was really showing teeth and growling and even making snapping motions at the puppy. But all of that without actually touching the puppy or hurting it in any way. The nonverbal communication was extremely aggressive, but the interaction was gentle.
Please take some time to watch videos on Youtube or your favorite video platform. You could use search terms like "dogs play or aggression?" or "aggressive body language dogs" or similar.
There are good and bad videos out there. The videos you want to watch are:
- Ideally made by trainers, because they not only show the behavior but also explain how you could react
- Showing actual videos of actual dogs. Infographics or cartoon animations are worthless because they cannot capture the nuances of natural behavior.
- Not list videos with numbers in the titles like "how to recognize these 10 signs of aggression" or "5 things you should / shouldn't do with your dog". These are just clickbait and not helpful most of the time.
Please have a look at this related answer that contains a list of 5 videos I recommend to learn dog body language and signs of aggression.