"Fake" animal farms to help FPS by keeping the amount of pets down are possible, but I cant think of a neat way to do this. Plants are easy, Seed in, Plant out. But animal farms... i cant use animals as reagents.
You can make it more realistic, for example, hay in, milk out.
But I'd argue against the animal farms. The problem is not how, but why. Trading works jut fine. Who need so many domestic animals that hurts FPS?
Every caravan brings more than enough cheese, leather, yarn, fresh heart, meat.
We can also buy meat from trading workshops, in case of starving.
The reason why we have too many animals is not about the product at all. It's about the animal itself. So an animal farm can't help much.
Someone, sometimes want an army of cragtooth boars, or ironclad horses for human.
Want cats to hunt vermin.
Want drake scales. I doubt human's ability in breeding drakes. It's better to buy the scales from dwarves.
Want eggs. Egg is not better than meat in any aspect. Animal farm has not the same sense of hen and nest box. So it's not useful and not fun.
Want pets to make people happy. You can write an reaction of "look at the beautiful birds", but it's better to actually look at some statues in the meeting hall.
So the best way of keeping the amount of pets down, is not to breed them at all. We don't need an alternative way to breed them, really.