For stuff like that, is it a bad idea to use some Loctite/Permatex Purple on the threads? I would imagine it will just vibrate off again the more it gets played.
LUTHIERS HATE THIS ONE SIMPLE TRICK!
Tons of acoustic guitars have screws/nuts like this, right? And it generally isn't a problem. I wouldn't jump straight to the threadlocker, but would reinstall with proper torque first. You want it tight, but not so much that you're crushing/damaging any of the wood involved.
But if you've got that nut nicely snug and it still loosens up over time, using a gentle threadlocker seems reasonable. I've got Vibra Tite VC3 and use it where I don't want to use the standard blue locker. The VC3 dries into a gel consistency, and they specifically describe it as vibration damping, adjustable, removable, and reusable. I haven't used the purple, but it seems like a good option.