(I ended up not using the dowel even though that is probably the best way to do it. Since my wood patch was 100% oak I thought the repair might be strong enough "as is". If my repair does not last I will have to go back and redo it the "correct" way.)
1) I cut the patch from the wood using the Dremel. Then I trimmed an area in the sewing machine board to match the patch. I coated both wood surfaces with wood glue and clamped.
Here are some pictures with the clamps removed.
2) Attached my repaired board to the hardware on the machine's cabinet
3) Screwed the machine itself back onto the cabinet's board, attached the belt, and treadled away
What a nice stitch this machine makes. I plan on doing some red work using this machine. Here is a closeup of the stitches.
Up until now, I was only able to stitch by manually turning the wheel. This treadling is MUCH nicer!
Thank you for all of your help Glenn! I really appreciate it.