Cutting a rabbit is the solution. When the glass goes in, you can use a strip of wood to hold the glass in. I wouldn't glue that strip in, in case the glass needs replacing some day. A pinner would do for that task. The strip might be something like 1/4 x 3/8. You will want to finish that strip and the doors before you put in the glass. Those Sommerfeld bits are really great, aren't they.
I bought the entire set of Sommerfeld videos. I find they have been among the most helpful videos I've seen so far, in part because they follow the project through completion and don't skip details like so many videos do. They are very coherent, each video seems connected to the others. And, no lame comments. I'm a fan and don't even get a toaster for it. LOL