re: Best mirrorless ILC for amateur cat photography?

Why mirrorless? For your use--"amateur cat photography at a shelter"--I don't see any advantage to a mirrorless camera over a DSLR. Given that, and the fact that budget is a concern, I'd recommend an older entry level DSLR from Canon or Nikon and a 50mm f1.8 lens.

My 2¢ - Mark