Some makers, even good ones, do glue the bridge down on top of the finish. This is most common when the finish is something like UV cure polyester, or epoxy, which are used less by high-end hand makers (although the polyester is gaining ground).
In terms of the repair, the only correct way to do it IMO is to remove the glue from the bridge bottom and re-shape the gluing surface, if need be. You also need to get the finish off the top, so that the bridge can glue down to the wood. This can be difficult, particularly with the hard modern finishes, which is one reason why they tend to glue the bridge down without cleaning the finish up. This is not a trivial repair; unless you have a fair amount of experience I'd find a local shop to do the work.