My strategy for preventing inbreeding is to turn contraceptives on for ALL of my animals at first. Then I selectively choose breeding pairs and turn their contraceptives off. Once they breed, I go back to the animals panel in the Zoo menu and turn their contraceptives on. Then once the babies are born, I immediately turn on their contraceptives as well.
In simpler terms, only the animals that I want to breed at that particular moment don't have contraceptives. So, I have no babies being born that I did not choose the parents for.
Once you turn the contraceptives on for all the animals at the beginning, it is pretty easy to manage.