This morning I rabbeted out the back of the wainscot frame to accept 1/4″ birch plywood. The frame is made from borax treated radiata pine, suitable for wet locations, and very resistant to Formosan termites.
The inside of the frame corners needed to be cut square to fit the plywood (alternately you can round the corners of the panels), using either a chisel or an oscillating tool. Once the corners are cleaned up, the panels can be cut to size and fitted from the back.
Since contraction and expansion of plywood panels is not an issue, I glued the panels in with waterproof glue, and pin nailed them to hold until the glue is dry.