I find that going on raw, you dont' really have to worry about the kind of meat you're feeding. As long as you pay attention to the total amount you're feeding. Average is 2-3% of body weight, so if too fat, then to for the lower end of the spectrum at 2%. Likewise if too thin then increase.
Yah, I'm one of those ppl who don't believe in sport hunting but I won't get into a debate here. Just wanted to say I agree that if the animal is suffering, you might as well help it along the way. It's very cruel to just leave it there.
But back to the OP, it was a pretty sad situation, but at least it's not suffering anymore and your kitties got a source of fresh meat.