Fixing a broken water bottle mount/rivet nut

I was removing my down tube water bottle cage for a race, and when re-installing it I broke the nut on the inside of the frame. I didn’t think I tightened it anymore than usual but, it seems I did. I’ve done some research and it looks like an easy fix with a wide-flange, L shaped, 5mm rivet nut. Park Tool has an example of how to fix it on their website.

I’d like to fix it myself rather than taking it to my LBS and having them ship it to the manufacturer and being without my tri-bike 5 weeks before my A race.

Since it is a carbon frame: 1) What is the best way to get the remaining piece of the rivet nut out of the frame? 2) Should I be worried about hurting the carbon when installing the new rivet nut?

Has anyone else had this problem before? Or should I just get a behind the seat bottle holder, and ship the bike off after my A race?