Stringers are structural members that run fore and aft as opposed to bulkheads that run port to starboard. Now that I think about it, I am not sure if there are stringers in the 208 that run all the way back to the transom. I kind of remember that if you look in the aft lockers, you will see platforms that hold the batteries on one side and the oil tank on the other and I think those platform don't extend to the transom. There is a gap between the two areas.
Can you post some details ( not pricing) of what the repair entails. If the core is rotten, most if not all of the transom skin and coring will have to be removed. Then a new core ( something like Coosa composite) would be epoxied and glassed in with several layers followed by more glass and resin to form a new transom. Part of the job would also be to fair the transom, sand of course and apply gel coat(color matched). The transom cap would probably be replaced as well. The cost and complexity of the job depends on whether the transom has to be removed all the way to the hull sides as well as up to the rub rail line.
I do have a question. On the photo of the swim platform side, there is a black line above the transducer area and above the platform. Is that line a crack or something else?