I only use pork fatback. I have old hunting buddies who have been using deer and elk fat in their grinds for 50+ years and never had a problem, but I don’t take the risk that it could go rancid.
I killed a black bear 2 years ago and it had tons of back fat, which I chucked. I am always reading that bear back fat can be fantastic for cooking, when it is rendered. If I am lucky enough to get another bear this season, I am going to try that.