After seeing the pic above, I'd say you may want to consider using a composite transom solution like, Arjay, Nida Bond or Seacast. You would then simply dig out all the old wood, mix the composite and pour in a new transom. This would make for a permanent fix with the very least structural damage to the rest of the hull and probably no welding. I'd then make up a new top cap to cover what is cut away at the top.
Here's some links on the stuff I refer to:
I have used Seacast and have heard good things about the other two brands as well. Seacast is the most expensive and the most popular but they were the first to get into the replacement transom market. Nida and Arjay I'm told are the same stuff, but can't confirm that. Both are used in several OEM boats from what I've been told.
We did a transom which held twin Johnson 250hp motors with Seacast and the result was a rock solid transom which so far is fine after two seasons of hard use. The boat was not mine, nor was it my project, but I prepped the transom for the pour.
The old wood can be dug out with a combination of drills, chainsaw, chisels, and a shop vac. If you can open up the entire top, an aluminum boat transom can often be removed whole depending on the degree of rot.