I would think to just remove the tooth particle wouldn't cost you anything. Insurance probably wouldn't cover it. As far as the problems with the tooth next to the extraction site this will need to be discussed at your consultation. I have a form in my office that I have my patients sign before their extractions. Here's a list of some of the risks:
Loss or loosening of dental restorations
Loss or injury to adjacent teeth and soft tissues
Sinus exposure (upper teeth)
The bolded is iffy. If the tooth was injured due to the process of the extraction then the dr may not be liable. I kind of see it as it was an oversight for him as far as the tooth particle, so he should be liable.
I'll be interested to know the outcome of this.