I know it was mentioned in the context of rabid Raccoons, but what about it just generally cropping up in rodents, and either being transmittable to deer and bear because a rabid animal "attacks" it, or they randomly get it very rarely.
I thought of it because we have those rats, and rats aren't normally that aggressive. Sure, you can die from the disease, of course, but it doesn't seem quite like rabies. You could add psychotic episodes, and/or one that made you involuntarily target nearby mobs sometimes, or attack at random without really targeting anything. (nearby mobs would include your townsmates.)
My thought is it could be very rare for beavers, and extremely rare for squirrels, at least in the light. In the darkness I feel like it would be more common. It's not a nice place.
Bats would have it relatively often, although the rats might not actually have it that often, unless it seemed appropriate. (they don't really get rabid that often, although get bitten enough by rats, and you would get rabies.)
It might not be especially relevant for bears, of course, other than just screwing you over with a nasty disease. Deer, bunnies, and stuff, it would make them actively aggressive. Nasty surprise when you are out hunting, keeps you on your toes (The deer, for sure. Minding your own business/fighting something else, a deer butts you in the ass and breaks your leg.)
Squirrels don't really attack, so maybe when you picked them up/wringed their neck? For that matter a number of animals you do that kind of thing to should maybe bite or peck you, either all the time, or some of the time. Like how you can lose blood picking thorns. Not necessarily infectiously of course. It might not also be unreasonable to require you to do something to restrain or more easily carry stuff like wild turkey and maybe even domestic turkey, who probably wouldn't like just being randomly carried around much. I feel like they would get wriggly if you put them in your bag. Again, it's a lobster thing, you band them so they don't get nasty, why not a turkey?
It always kind of seemed strange that they resign themselves to their fate when you ko them. I can see them maybe staying ko'd for a bit, but after a while/out of the coop? No.
Stick them in a sack or tie them up with fiber or something around the legs/bill. If not they would thrash and claw and peck and do minor damage.