I knew someone who flew the FAA's Sabres for years on flight check missions. This is extremely specialized flying. I learned that the openings are few and far between. Very experienced people are hired. I suspect one needs a connection to get these jobs; I don't think the FAA throws open a register to solicit applicants because it doesn't need to.
No offense, but you don't have enough time or experience. Just the same, first thing I'd do would be to go wherever federal job openings are posted and look for a listing. Then, you might check directly with the FAA in OKC; I'm not sure exactly where at the Monroney Center to call.
Sorry to paint such a bleak picture. Believe it or not, getting a job with a major is easier than getting a Flight Check job. Lots of luck to you.
I know of a gentleman who secured a Flight Check slot up in the NE region within the past several years (during the peak of the "pilot shortage" ) who had between 2-3000 hours, all of it piston. Explore your options.