I have had same experiences….my birds do not care for dead mice either, I think they prefer the thrill of the hunt lol
But my barn cat on the other hand likes 'em dead or alive :) & he has even eaten mice after they took 'just one bite' & bit the dust, the poison in a dead mouse is not enough to do in a larger mammal like a cat or dog I do not believe, but IF they ingest the mouse poison directly & in large enough dosage than, yes, it can kill them too.
I lost a cat one time after he walked on some granular mouse poison that I did not even know he had gotten into, he licked it off his paws, was sick for 2 days & then had to be put down because the vet said there was nothing he could do to save him
….now I never ever use the bait packs, I use the bars only & like Doug says, always put them inside pet proof bait boxes….
PS, this works equally well for rats….. I had the joy of my first encounter with rats on my farm just this last simmer…..ick!! 