Lyman,
Somebody like "ExPonent, Failure Analysis Associates", in Menlo Park, California.
And,
Who would you choose?
It is a small world out there in the accident investigating field. Most private firms wouldn't have the capability to do a complex investigation on their own. Exponent does have a person knowledgable in transportation investigations according to their web page:
Dr. Dickinson specializes in transportation related issues, particularly accident reconstruction and safety analysis of transportation related products, systems and installations. He has had extensive experience spearheading many investigations of major transportation related accidents and failures, and his current research focuses on urban transport studies, safety analysis of railroad equipment and operations, evaluation of aviation safety and air traffic control. Dr. Dickinson is a former member of the National Transportation Safety Board and held several engineering and research positions before joining Exponent.
Exponent was used by the NTSB in the investigations of TWA800 and AA587. They were also used in the Turkish Airlines 981 accident investigation lead by a French Commission of Inquiry appointed by the Minister for Transportation.
Personally, I am generally satisfied with the way the NTSB does their investigations and don't see the advantage of taking this function private. OTOH, I think the FAA could do a better job when given recommendations by the NTSB.