Answer:
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.