Figured I'd respond to this old thread, as I just had the same exact issue and easily fixed it at home in about 30 min.
My insert face (Sigma Doon) was popping out in the exact same spot as originally posted. I used a mini flat head screwdriver and worked around the edges of the insert face to get it off completely. One it was off it was clear that it wasn't exactly flat anymore from prying it off.
First I cleaned the thin film of hot glue off the back of the thin face plate that was removed.
Next I put it back on the putter face, keeping face to the sky so it wouldn't fall off, and heated it with a hair dryer until it was hot to the touch. This made the thin face plate go back to its original form...perfectly flat.
Lastly I put it back on with a very thin film of epoxy and good to go. No need to clamp plate down for epoxy either. It was so perfectly flat I just set putter down on back of face and set it on there and let it cure...the plate sits in its housing perfectly on its own.
The only thing I lost along the way was the really thin black gasket that was between the face plate and putter body. It deformed when removing the face but didn't have any function besides cosmetics. Putter face looks just as clean without it...can't tell it's gone since it's so thin anyway.
Pics are of putter after repair.
