Nice repair Oliver. I love it when someone suggests a solution to a problem. I just finished my first picture frame for my wife. I made a mistake and attached a half inch thick piece to the underside of the frame so the painting would fit. Unfortunately, it was not completely flat and there were some gaps maybe 1/32nd wide. Hardly noticeable. But when I glued up the frame, that small gap showed up as mismatch of the corners of the face.
Careful shaving with a chisel and judicious sanding covered up the problem. But the lesson is clear, do the frame front first, THEN glue on the piece for the rebate. Someone on here has a signature line about the expert is the one who knows how to cover a mistake.
Now comes finishing.
Man, this woodworking stuff is great.
The more I do, the less I accomplish.