I do not know what a "negotiator" is in the context you are using it, and I have been practicing in NY for 33 years. You should have an attorney who is knowledgeable about, and skilled in, products liability cases. You do not indicate if you have healed well or not, but if you have permanent damage negotiation will probably not be sufficient.
Its a little late for the realization that you need an attorney. You did your child a disservice by trying to handle this yourself. Depending on the small claims rules in your state, it might be possible to get a trial de novo.
The reason is because there are parents out there who would abuse their fiduciary responsibility to their child and take all there money and the only way to protect children against such actions is to only authorize its release in extraordinary circumstances.
This is a very good question. I can only speak to what my office has always done in these situations. We have always tried to fix the amount of the lien BEFORE finalizing the settlement for various reasons. First, I believe the client has a right to know this information when making a final decision on whether to settle or not. Second, I believe that the is more leverage with the lien holders before the settlement, in that there is still the risk that they won't get paid if the case goes to...
You do not need a pro bono attorney. Dental malpractice attorneys work on contingency fees. They get paid out of the proceeds and advance the costs. I hope he has malpractice insurance because, if not, he will probably file for bankruptcy because I would expect that he committed malpractice on the majority of his patients. This is an all too common scenario. Consult with a local dental malpractice attorney and good luck.
If there is a chance that the insurance company will not offer what is necessary to resolve this case then you might want to consider hiring an attorney to protect you personally. Some insurance companies, although they won't tell you will protect you in the event of an excess judgement.