Mark, many years ago I used to do stained glass. A good stained glass piece will have its own frame to hold every thing together. Your frame should only be for decoration and should not have to add any support what so ever. What you will need to do is "securely" attach the top of the frame you build to the top of the stained glass frame. Technically you should not need sides or a bottom to your frame.
Having said that if you want the stained glass to float inside your frame I would biscuit the mitered corners and then drill 2 holes in the top and 2 in the bottom into the corners. Counter sink and plug the holes.