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Welcome. Nice clock. The look is very interesting.
I think I'd want to make a jig for the router to ride on, with the curved piece clamped down underneath.

I would want the "donut" larger piece to have an opening straight through, and then a smaller piece with the tweed wrapped around it tightly and glued in place. This smaller piece would be thicker, and cut slightly smaller than the opening it fits into to allow for the thickness of the tweed. The thickness would hold the the clock works and leave a spot for a keyhole to hang it.

If I wanted to replicate the curve of the stave, I'd cut it out of a second stave with the identical curveature. I'd cut the opening for the clock mechanism through the smaller piece using a router, but I'd drill out most of the opening to reduce the amount of work the bit had to do.

The jig would be a little like an exact fit jig for cutting dados, but as wide as the cut you want to make. For the second piece, I'd lay the top opening on the smal piece stock and pencil in the size as a cutting guide. I'd also drill out most of the top piece and use a jig saw to cut pretty close to the line for the top piece. Then put it in the jig and use a long trim bit to smooth out the cut. You could use a chisel to square up the corners, or leave them rounded because you'll mark the small piece through the larger one's opening,

I'd probably use a band saw to cut the smaller piece to size, and use sandpaper to size it just right. I personally would use a flat, thick piece of wood for the insert part so I could put a keyhole in the back to hang the clock.

This is how I'd approach it, anyhow.
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