Bill,
Many years ago, I picked up a Gibson bass with a poorly painted finish on it. I stripped it down to the mahogany and put a clear finish on it. The mahogany really cleaned up nicely, and the bass looked great when it was done.
If you didn't pay too much for the SG, as you said, you could try to do the refinish yourself. If it doesn't meet your standards, you could take it to a pro, and let them clean it up. If you think you are up to the task, I would think it would be a fun project. You may find out you have some hidden talents you didn't know about. Good luck, and please put up some pictures.
Regards,
Pete