I know it's easy to imagine that this is in fact the case.  I also know
you've had your problems personally with dentists.  I don't doubt that
you have been treated poorly, and can't rule out the possibility that
you have been treated negligently.
    Of course, there MAY have been negligence in the OP's procedure that
contributed to the file breaking.  But that will be exceedingly
difficult to demonstrate, and every dentist that does root canals has
instrument failure from time to time.  In other words a broken
instrument does NOT equal negligence.
    I might add that many dentists would not even mention the broken
instrument, and in many cases it would be difficult to know that it
happened unless the dentist (or another dentist) told the patient about
it.  The majority of cases with broken files in fact succeed, and many
without broken files sometimes fail.  Please note that I am not
advocating breaking files in canals.  I'm just giving you my impression
that from the OP's description, the dentist sounds to me like a stand-up
kinda guy (and believe me, if you read my posts regularly I have no
problem saying so when I think this is NOT the case).
    I do get the nagging feeling that part of the money story isn't
here--perhaps the $245 was an insurance copay and not the total fee.
It's unclear who finished the root canal, and exactly what the
endodontist charged the patient for.
    Finally, while it may sound callous and inconsiderate to say things
like this DO happen, and it is tempting when things go wrong to want to
blame someone, I can guarantee that even the very best doctors (and
maybe especially the very best doctors, since they are often entrusted
with the most seriously ill patients) have poor results.  Insisting that
doctors conduct themselves competently is one thing; assuming a bad
result must be the doctor's fault is quite another thing.  Even the
lawyers recognize that for a condition to be actionable, bad results is
not enough.  You have to demonstrate negligence (deviation from
generally accepted good practice), damages (medical, financial, etc.)
and proximate cause--that negligent action directly caused the damage to
the patient.  We do not have that here, because we would have to
demonstrate that the breaking of the instrument was the result of
negligent use.  You're not going to do this.
    Since I believe any dentist will understand that many patients in the
OP's position will be angry, I think the OP was totally correct in
bringing this issue up with the dentist.  It is quite possible that a
mutually agreeable solution will be found.

Steve