Agreed, with weight not being a concern, you can do pretty much whatever you want.
I haven't personally tried it, but I don't see an issue with, say, 40psi foam. That's orders of magnitude more resistance than your camper has pressure, but agin I have t done it and I could see some nuances to it maybe.
Sounds like you load and unload the camper a lot. Do those have a fiberglass bottom, or could you attach your plywood to the bottom of the camper?
Regarding warping, yes cheap CDX plywood will warp. 3/4" BB or MDO will not warp and even if you kept it wet for a few years straight, it wouldn't rot.