I've been using 3M ATG Transfer Tape for years on leather arrow strike plates and shelves. It's similar to the 'flooring tape' sold in the home improvement stores. The 3M tape is top shelf stuff. It's crazy sticky, holds through all sorts of weather, and I've had it on my hunting bows for 15 years and it's still right where I put it. If you want it off, it comes off without affecting the finish.... well, on any finishes I've used so far. Often, you can just roll/push it off with your thumb, or if you're in more of a hurry, use a moderate solvent to soften it, then wipe it off.
I'm confident it would hold thin rawhide on there, although, I wonder how bad the rawhide would want to curl and pull away as it was drying after it got soaked in the rain.
I bought the stuff at Lowes once after I ran out of the 3M, it worked ok. Now I'm stocked up on 3M ATG.