Here's what they look like now:
When the painting was finished, I traced each lid on the wrong side of the paper I wanted to use and cut on the traced lines. To apply the paper, I used Scotch double-sided roller adhesive, which is easy to use, very thin, and holds extremely well. I ran the roller around all of the edges of the lid and then in a lattice pattern over the middle, then carefully aligned the paper and smoothed it on. So far I've had no problems with the paper or the paint coming off, but you could go over both with a craft sealer to increase durability.
I'm really pleased with how this project turned out, especially since I'm pretty inexperienced with painting. There's often a big difference between how you envision a project and the final result, but this project ended up looking exactly as I hoped it would!