The idea that jars me the most is that gryphons are already a hybrid species by definition - what's more, they're a hybrid of two existing BK species, the Puma and the Roc. Introducing a hybrid breeding system without addressing that would be hugely inconsistent, but that would mean a complete overhaul of the species - if inventing new hippogriff genetics based on unicorn and gryphon genetics sounds hard, imagine reconciling existing gryphon genes with the puma and the roc. Nightmare. D:
Other than that, however, I do like the hybrid idea. If we follow mammal vs. reptile and herbivore vs. carnivore laws, and set aside alien species like the Symurgh and Throdama, we are left with Dragon x Cockatrice, Puma x Roc, and Unicorn all alone. (Man, I hope we get another herbivore species some day!) Puma x Roc would be a Gryphon, as mentioned before. Gotta set that aside. Considering the cockatrice...it's so fundamentally similar to the dragon. What would a hybrid even look like?
The best I got is some sort of semidragon - retaining the long neck and tail of the dragon while inheriting birdlike qualities from the 'trice, ultimately resembling a swan or maybe a flying serpent. Gryphons could be explained in-universe as having developed ages ago from horses and big cats, perhaps due to some powerful magic, but they're genetically independent of the original species or something. The reptile hybrid could be more like a mule IRL, in between a horse and donkey, bred by Beastlords and unable to reproduce.
Thinking about genetic inheritance is making my head hurt...all I can think is that predefined combinations of unique genes (Fire Gland from dragon mom and Acid Gland from cockatrice dad) might result in some sort of species-specific combo attack, like Acid Shot. Mebbe.
*~*Combat Unicorns*~*Deviant Rocs*~*Silver&Gold Cockatrices*~*