My mother's boat had a similar thing going on a few years ago. I removed the engine and verified the wood was still in good shape by drilling multiple holes and inspecting. There was no flex in the transom, either. I contemplated skinning the outside of the transom to rebuild, but decided on a different approach as I was reasonably sure the transom core was fine.
I infused resin into each hole I had drilled to seal it up. I then bought 1/4" SS plates and created multiple "backer plates" for both the inside and outside of the hull to be double sure there would be no issues with structural integrity. I cut two large squares (maybe 10") for the outside and two wide rectangles (from memory, roughly 3"x24" or 30") for the inside. One rectangle was used on the inside of the transom lip where the the motor mounts and the other rectangle was used further down the transom on the inside. Thru bolts were used to shore everything up.
As mentioned, the bang cap has absolutely nothing to do with structural integrity. You would need to fab a bang plate that is U-shaped and thru-bolt it through the transom. But that may not be enough.
Based on the info you have provided, though, it's hard to say what you will need to do to fix it. Best bet, if this is not something you're comfortable with, is to get it checked out by a reputable repair facility.