TL;DR - way too much possible crossing between categories.

A few points of criticism:
1. The loon is certainly not a songbird. Not by taxonomy, and not by vocalization. They call rather distinctly, but I don't think you'd deem it a song.
2. If taxonomy, then, is not the criteria for the songbirds list, then easily the p(ewe)e could swap with the ta(nag)er within the animal name category.
3. King(fish)er feels like a cheap answer within the animal name category, as it is by that very animal the bird was named. It's not as though the word is "hidden" in there.
4. A simple search of IMDb will yield film titles with many these birds (Loon, Pigeon, Cardinal) so the Oscar nomination is probably an important distinction to make the category solid - as (kit)e is another animal name, and the cuckoo, while not scientifically a songbird, also has a distinct call like the loon.