Oh man, I hate cavities. I really hope I never have to have another tooth filled.
There are far worse fates for teeth than simply requiring a filling. It's very disappointing to see a tooth that could have been saved with a filling a year earlier require a root canal, surgical crown lengthening, a post and core buildup and a crown - all to be completed with a questionable final prognosis probably ultimately ending up in an extraction, socket preservation bone graft, and implant placement a couple more years down the road.
To separate teeth that are bonded together a dentist might use something like an ultrathin saw.
In TheFountain's case where there is already a space between the teeth but there is a catch I would use a thin mylar sanding strip
. If I had the piece stuck in my mouth and didn't have access to such luxuries I would probably attempt to use another piece of floss to get the first one out. If the catch on the filling is very thin it might be possible to snap it off by this method. If it is significantly more bulky then that will not be possible and the tweezers or a dental pick might be of more help.
However the office should be able to get you in for ~10 minutes to get the floss out and sand the interproximal surfaces smooth. If they're a dental mill as you've hinted you may have to go there on standby.