Many times dry rot can be repaired by getting the wood completely dry, then using epoxy to rebuild it. Do you think that might be possible on yours? The photos are not clear enough to show the actual soft spot(s) so how you go about it is up to you. Use a heat gun to remove the PVC strips, and all attachment points. Lots of heat, but keep the heat gun moving. The last thing you want to do is melt the PVC. It can take a few minutes of heating just to be able to remove a small section. If the material starts tearing, it is either bad or you aren't doing it right.
The glue needs to be the proper PVC two part glue and temperature and humidity are critical for adhesion.
Originally Posted by Nasher
Using Acetone on PVC before the first layer of glue will soften the surface and allow the glue to seep into the top layer giving really good adhesion.
Actually that would be MEK to soften and prep the PVC for bonding. Acetone is used for cleaning the old glue off, as it is not as harsh as MEK.
I found that Scuffy Pad works excellent for removing glue without taking hardly any material. The courser the Scuffy Pad the more glue it can hold inside itself. Sanding is a no no, and all the old glue has to go. Good neoprene gloves and a respirator are in order along with doing it outside in open air.
Building a new transom can be done fairly easily with glue, weight, and a router with the right bit to follow the old transom as a pattern. Screw them together or double stick tape and just use the old transom as a template with a bit that has a bearing. You will need to use a decent quality marine type plywood. I doubt you can buy the right thickness so you will need to buy 1/2" ply and glue two sheets together, with waterproof glue.