I ended up unscrewing the mounting hardware, taping together 4 layers of foam-core board (y'know, the kind you use for science fair presentations in elementary school!) in roughly the right shape, wrapping a cut-open ziplock around to make it fairly water-proof, and screwing the mount back on over the foam spacer!

It doesn't look super sleek, but it works! The only problem is that the new screws I used (the existing ones were too short) stick out a little in the bag. I'll just have to be careful not to snag my hand...

Thanks for all the great ideas!