Thanks again for the responses.
The doors (not yet sanded or finished) are flat, but leaving them as-is is not an option. To answer Pottz about the weather, in the two weeks since I started the shop (garage) work, temps have varied from 30 to 80 degrees, and humidity from 50% to 90%. The doors have stayed flat because I’ve been keeping them sandwiched between other panels when I not actively working on them. They will cup if I leave them alone (and I will finish both sides evenly).
It sounds like wood battens are the closest to a consensus opinion here. I may go that route, but I’m leaning toward the original plan, mainly because I’m curious about it.