As with many queries involving Choshen Mishpot, the answer to this particular questions depends on a number of factors. Firstly, we would need to know if the dentist is really responsible for what happened. Then, was it due to negligence or was it unavoidable? Further, is a dentist considered to be in the same class as a doctor, who may be absolved of responsibility under certain circumstances (as far as the Halachah is concerned) if he wasn't negligent? Did the dentist charge more than the normal rate (which might oblige him with extra responsibility)?

With regard to suing in court, in general it is forbidden to sue a Jew in Court. One must sue in Beis Din first. If he refuses to attend, Beis Din may give permission to sue in Court.

However, if one is suing the gentile-owned insurance company rather than the dentist himself, it may be permissible to sue them directly in court.  All in all ,then, in this case one should apply to sue in a Beis Din, and follow their instructions.