If you can attach a square block to the end of your dowel, that will keep it from rolling on you. Then like Kevin said, just make some relief cuts to eat out a flat recess. I'm thinking that you'd use a table saw. Could be done on a band saw or scroll saw as well or even a table router.
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Thank you both! Very helpful info. Here is where I am at so far (;
Any tips on the best way to notch the dowel so the lid can fit into it?