If she WASN'T an off-duty cop, then yes, she'd be guilty of impersonating an officer... POSSIBLY of Disorderly Conduct, as well, depending on words and phrases used. The two of them could get cozy in the back of a wagon together.
If she WAS an off-duty officer, then yes, producing the badge and informing the hit-and-run driver that she was under arrest would be an option. More likely, she would call it into the dispatch center, identify herself by name and payroll/badge number, and advise of vehicle description, tag, direction, and incident, and await a uniformed officer to make a traffic stop. At that time, once the officer has the suspect's information, and has verified that there's no weapons in the car, she could let him know what happened. Officers can make warrantless arrests based on the testimony of another officer, provided the incident had just occurred.
Also, after conferring with my Sgt, my figures from earlier were roughly correct, if a touch conservative. A good shoot, after factoring in all the call-out pay, overtime, and straight time for EVERYONE involved, can run about $10 grand. For a BAD shoot, you have to start including people like the DA's office, and that racks up the price tag.