How do I approach cutting a new marquetry inlay to fit the space in a metal bracelet
I think that a great way to transfer the shape to the wood would be to use a rubbing to create an outline of the space which you can then cut out of the paper and transfer to the wood. This is sort of reminiscent of how Indiana Jones saved the instructions for finding the grail from the shield under the library:



You can use either a crayon or chalk or pastel or some similar flat-sided medium and a piece of thin paper like tracing paper or tissue paper. 


Place the paper on the side of the bracelet where the opening is and hold it in place firmly.
Rub the crayon on its flat side along the paper.


The high points on the bracelet should be colored in and the relief area you're trying to fill with wood should stay either white or be much lighter in color.

If you've held it still enough, you should have the exact shape you need. At this point, you might want to make a copy or two of your rubbing so that you can cut it out of slightly thicker paper and preserve the rubbing so you don't have to redo it.


Cut out the white area and test it to see if it's a good fit inside the space (it should be).
Use the cut-out as a pattern to trace onto your wood.


Be certain to cut it out slightly larger than necessary so that you can allow for final shaping to be done as you're sanding after cutting out the shape.

If you take your time, you should get an excellent fit out of your wood inlay.