If it was me I'd attach the carpet to plywood then screw the plywood to the doors. That way you can always remove them if you have to. Considering the moisture problems you've talked about I'd advise against glueing anything that might get wet in case you end up having to remove it to dry it out.

My plan is to use cover my doors with a rigid foam/plywood/laminate sandwich. Laminate facing into the van of course. Then secure it to the door using either sheet metal screws or riv-nuts and screws