The transom wood can be replaced in a number of ways. The object, though, is to seal it against water intrusion as best you can before mounting it in the hull. I use epoxy. (Woodonglass has a formula that does the trick well, and more cheaply.) You don't need marine plywood. Get some decent A/C, and use a couple of sheets of 3/4 thickness (typical) sandwiched together. You can use epoxy or resin or 5200 to sandwich the sheets. I combine an adhesive and drywall screws (stainless wood screws if you like) -- seal in the screws. When it's all put together, you seal the whole piece -- special attention to the edges. It doesn't hurt to seal the edges with the resin and polyester you are using (I use thickened epoxy.) Let the wood soak up the resin/epoxy, then coat it.
For installation, use the adhesive of choice. I've used 5200 and that works well, and gives some set up time. Dry fit first. I also use screws again from either the inner or outer skin, or both, then seal in the screw heads. That requires painting or gel coating for finish, so depends on your plans. Another approach for fasteners might be carriage bolts, through bolted from the outside. In combination with the other hardware (e.g., transom eyes), that can give a mechanical method to go with the adhesive.
Water gets into the transom core from the top -- little screws that hold the cap, for example. 5200 is the best for sealing up at the top (not permanent).
You have the hull opened up and might be able to get the old wood out of the envelope from the top. You will likely need to some up with some creative tools. I have an old crow bar with a sharpened end for gouging it out. I think I would try it from the top first. I've cut off the inner or outer skins too -- can do either one.
When finished, the transom will be at least as good as when it left the factory -- good for another 30 or 40 years.
