The door has a thick piece of wood under the rat fur. And yes, it can rot from
leaks around the window, hinges, and door latch.
First thing I would do is remove the door latch completely, inside and outside, and probe with a sharp instrument to assess how much rot you have. Check below the window and around the hinges as well. How’s the door fit, BTW? Any large gaps?
It could be simply that the screw holes have gotten wallowed out from time and vibration, and there are ways to fill and strengthen them so screws will hold. That’s the best case scenario.
If the rot is more widespread you’ve got some work ahead. I don’t think bypassing the problem by punching rivets through the outer shell is the way to go. If the rot is localized around the door handle, it might be a reason to scrap the old-style bread truck latch for a modern RV latch. You might have to peel back the lining and patch in a new section of core material,.
But it all depends on how widespread the rot is. Let us know what you find.