I think doing wood cabinets will be very nice. I built the cabinets for our house using 3/4" maple, and Maple hardwoods, totally glued, no nails anywhere. Way to heavy for your Casita
. As Floyd mentioned, our cabinets in the Casita are mostly riveted to the shell. We visited the Escape
factory in Canada. They have wood cabinets and, as I recall, epoxied blocks to the inside of the shell for fastening the cabinets.
I think birch is a good choice. I made a dresser out of alder with 1/4" birch panels and it is still is service 16 years later. Using panels insert could reduce the weight
but might not work well with all of the curves.
Remember that the interior cabinets ad strength to the shell so fastening is important. That said, I would not fasten them too tight since the trailer needs to flex while traveling.
OK, enough analysis. I would epoxy blocks at strategic places and consider using existing rivet holes with screws (or bolts) as Floyd suggests. When installing the cabinets, leave the holes a little larger than the screws to allow for some movement.