If you pick up gauze pads rather than gauze wrap, you shouldn't have any real problem with sticking. Just be gentle when you pull it off. From what I'm picturing, 3x3 pads should work well, right? The pads unfold, so you should be able to get a good covering. You can anchor it with either the cloth medical tape or, my preference, transpore tape. Run a piece from your wrist up over the thumb and back down the palm side of the thumb onto the pad of your hand, then use another piece around the wrist to anchor the first piece. Transpore and guaze pads should be availalbe at any walgreens. 4x4 pads are the most common, and will work fine.
Is it just the tip of your thumb, or is it bigger?
Edit; I can't describe this well. When I get home, I'll try to take a couple of pics of a wrap job and post them.
Oh, hydrogen peroxide and such kills viable tissue. With a dog bite, I assume you're on antibiotics?