If you use plywood, you have to use marine grade. That's because it uses non-water soluble glue. If you can't find smaller pieces of marine ply (my local boatyard will supply pieces to size..) you could always use a hardwood - Teak, Elm, whatever. You might find a good lumber supplier who will cut it to your size.
I recently added an Elm reinforcement to my dinghy transom. It's as solid as a rock.
If you need to join two sheets to make it thicker, I'd use West System epoxy. Get plenty in there to keep the water out and clamp it while it sets. I also used several layers of epoxy, brushed on, on the outside of the wood to keep water out. Then you can either varnish or paint. You must do one or the other as the epoxy is not UV resistant, so the varnish or paint keeps the UV out.
What I did was use some off-the-shelf Elm 2ft x 3 inch x 1 inch pieces, placing one layer vertically and one layer horizontally, using epoxy to bond the whole lot together, then bandsaw to shape. Yes I made my own plywood! Well it was inexpensive, and it looks great.