It might sound awful but your cat does what his instincts tell him.
A rat is a dangerous creature. It's smart and can bite quite severely. Attacking it from behind, weakening it, and making it almost die from blood loss is a safe approach for a cat.
A bird is also a dangerous creature. It has claws and a hard beak that can do a lot of harm. It can fly (or at least try to). Again, weakening it is the strategy to use.
We are against such approaches but these are natural mechanisms. Also, a cat's instincts work regardless if it is hungry or satiated. A satiated cat can actually be more deadly since it has sufficient energy to hunt efficiently.
This doesn't mean that a cat will treat other cats as prey. That's not what cats do so other instincts should be at work here. The kitten will act as a cat does (especially if it is grown enough) so the two will be able to communicate in the same "language".
Anyway your cat might want to show the kitten its place in the herd so the introduction of a cat at home should be gradual. Start with scents. Then your two cats should be physically separated (but hear, smell and preferably see each other). A large cage for the kitten is a good idea. It can stay in such case for weeks without much harm. Socialise the cats with each other only when you have full control over the situation (and do not overestimate your capabilities - cats are extremely agile and smart too so they can fool you). When the older cat no longer poses a threat to the kitten you might start leaving them alone (again, at first just mock it and monitor situation), then gradually leave them together more and more unattended.
In general having a second cat is a good idea. It channels some of older cat's energy (sometimes fights might look violent but that's also normal, you have to learn how to recognize if cats are playing or really fighting).
As for age, 3 weeks is way too little for a cat to be taken away from its mother. Unless there are serious reasons to do otherwise (the mother was killed, has a high risk infection or is aggressive towards its litter or specific kittens) the kittens should stay with mother until they are 12 weeks old. 8 weeks is absolute minimum.
At the age of 12 weeks kittens no longer psychologically suffer from being detached from their mother. If it happens earlier, they may suffer from orphan disease, negatively affecting their behaviour (among other things increasing a risk of aggression).
At the age of 8 weeks kitten are fully transferred to eat solid food and do not need a supply of mother's milk. Taking kittens earlier negatively impacts their immune system and at earlier stage it is a life threat as you can hardly replace the mother's milk with any substitute. Specifically don't try using a cow milk, it can cause diarrhea and lead do dehydration and death of a kitten. Moreover such small kittens should be fed every 2 hours. Can you do that (at night as well?)
The earlier kittens are detached from their mother the larger are the negative effects. Be a responsible pet owner, resist temptation to have a nice cuddly kitten and take it until it's really ready. If you really want to take a smaller kitten, adopt one that really needs it (for reasons given above). Still I would avoid taking kittens smaller than 5-6 weeks unless you really know what you're doing.
So please, rethink the idea of taking so small kitten.