A quarter at 20 yards is quite a feat. I hope you make it.

I have never worn gloves while dressing any game. I was cut once dressing an elk by an old grandpa that was struggling to hold up a hind leg while I was working on the paunch. So I gave him my knife and grabbed the leg, and wouldn't you know it, he drug the edge across my thumb knuckle and opened me up. So I have elk blood running in my veins. Doesn't seem to have made much difference.

For things like tularemia, there's an effective method for opening up the paunch on a rabbit and spinning him round in a manner that causes all his innards to expel away in a balled up mass. Thus removed, the carcass is fairly safe to handle, though I would still exercise caution with open wound exposure and quick cleanup after further dressing.

I won't handle sick critters. If they don't act proper beforehand, I won't shoot them for harvest. I might pop a peculiar varmint if they look loco. Likely as not they are rabid. I don't go after prarie dog or marmot because they have are a known vector for the plague, and who needs that? Tree squirrels are fairly easy to figure if they are safe or not. Either they act squirrely, as they should, or they act noticeably impaired. You can put them out of their misery, but give the corpse a wide berth afterwards.

In short, I am not too worried about exposure. As far as I know, no one has yet contracted chronic wasting disease. In the case of getting my thumb slit, a rubber glove wouldn't have helped me in the least.