Can a Part 135 operator use a contract person as a Check Airman?  Or does the Check Airman have to be an employee of the Part 135 operator with their name in the OpSpecs?

as long as they meet the requirements. Who cares how they get paid. Check airmen are not in the op specs.

Not uncommon for small operators who can't get a ride with FAA to do this. You'll need their POI's approval and will have to submit your info to them, on top of the company looking for the ride to make a formal request. Also a good idea to let your POI know that you're doing it because you'll have to submit it at end of year regardless.

FlightSafety has checkairman for part 135 companies.  Just have to be approved by the POI and have an LOA. Also have to be an approved contract training provider on Opspec AO31 to provide training.

We at Helicopter Institute offer Contract Check Pilot services and A031 Contract Training to our clients in Bell 206, 407, and AS350 aircraft. Unlike FlightSafety or OEM Part 142's...A031 Contract Training and Checking provided by an operator provides a path to complete the entire .293/.297/.299, not just the aircraft-only portions of the check. Much better solution so the customer doesn't have to complete additional flight checks outside of the training/checking agreement.