I did exactly this mod with a Tissot bracelet for exactly the same reason. I used a 0.9mm PCB drill in a Dremel with a tiny drop of oil and VERY light pressure. The speed was probably a little high but it did work. Getting the hole started in the right place was the trickiest part but I probably could have made a more defined centre punch first. As mentioned, drills of this size and type are very fragile so I knew I was taking a risk, but worst case I thought I'd end up with a small mark from a partial hole that wasn't successful. Would be happy to give it a go with yours if you accept the same risk. Drop me a PM if you'd like.
The 4th hole (on the left) is the one that I added
(As an experiment this photo is set to "Private" rather than "Public" on Flickr so please let me know if it's not visible. I can see it fine but not sure if it it'll work for everyone)