What I like about Tom's answer is the focus on goals to help in decision making.
The maximal repair for this problem has been performed recently by a DSII owner who cut off the entire rear of his boat, rebuilt the transom and then glued it back together. For someone who is organized, reasonably skillful in a workshop and persistent in seeing this through, it's doable that way. The result would be "good as new". I personally might shy away from something that big, mainly because it seems daunting. It's probably true that the fiberglass work required is way less demanding in terms of skills and practice than doing equivalent work on a wooden boat. Anyway, that approach is written up somewhere.
I seem to recall reading about someone just slicing off the fiberglass skin to get access and then rebonding that after digging out the rotted wood. If you can get the rear face off in one piece, you can probably reuse it, and the extent of fiberglass work might be limited to "curing" the cuts you made.
In contrast to the DSII, on a DS1 the transom is a single layer of laminate, with a strip of wood backing only where the rudder is attached. Because the DS1 has about 18" of rear deck (also single layer of laminate), the transom is effectively an L shape, which makes it a bit stronger and doesn't allow the actual transom to flex. For the DSII this design was changed to a sandwich, using wood as core. The wood being incompressible (when not rotted) makes it ideal for mounting rudder, motor brackets, swim ladders and all the other kinds of attachments you'll find at the rear of these boats.
Purely for stiffness (and to keep the sides of the hull in position) a foam core would have worked just as well, presumably. Panel stiffness is determined by the strength of the skins and their separation (core thickness). That's why foam cores, or balsa cores work. And for narrow panels the side walls provide the separation, that's why you see things like stringers that are simply hollow with rectangular profiles.
Even with rot, the transom "sandwich" will likely be strong enough to keep the water out and the hull sides in place (in moderate conditions).
But your rudder mounts won't be as strong and you certainly won't want to mount new hardware. And by drilling holes, you've committed to doing something - if only to close these against further water ingress.
In conclusion, the issue is fixable with a bit of mostly sweat equity. However, any decision to do something like go for trial sail before you commit to fixing this more permanently would be entirely up to your own judgement and you would be well advised to consider how you could temporarily shore up transom and rudder mounts to reduce the risk of any adverse outcomes.