Best Bear:
Sun bear by far. They are my favourite bear IRL, but I really love all species. In PZ, however, most base game bears have an awful potato shape without neck that makes them almost unplayable for me (the panda is a bit better because they do have a rounder shape). Then the polar bear has a really good model, but their not so realistic snow and space requirements put them behind the versatile sun bear.
Best Bird:
Southern cassowary, just because I love them IRL. But this is a sad category, all the birds in the game shouldn't fit in only one, there should be waaaaay more species. I like very one of the birds in the game just because of their scarcity, just like every individual of a critically endangered species.
Best Rhino:
Indian Rhino. Every PZ rhino has its weak points, so in case of doubt I prefer to give credit to base game, because I bet almost all the winners will be from DLCs.