You could easily make a fixture that aligns the end of your Dremel's flex shaft to drill the precise holes. It could be as simple as a grooved guide block that you tape or clamp in place while you slide the flex shaft along the groove to drill the hole.
If all else fails, here are a few more possible solutions to consider (though I'm not sure if this would have an impact on the sound quality since you're talking about a musical instrument, but here are a few more possible solutions to consider if the flexible drill and offset adapter solutions turn out to be impractical):
- Drill all the way through from the other side (from the left, in your picture--assuming that's possible), then plug the hole from that side using wood filler or a dowel.
- Cut off the part that will have the hole, drill the hole, then glue the part back on. (The cut would be a horizontal line in your diagram.)
- Similar to #2, cut off the very top part, use a router to carve out the channel (instead of drilling it), then glue the part back on.