243 to 6mm ?
The shoulder diameter of the .243 Winchester is .455 inches.
The sholulder diameter of the 6mm Remington is .431 inches. Although the shoulder of the 6mm Remington is slightly forward, there is too much diameter to clean up the chamber. You will only end up with a two shouldered case.
Also, there is a question of magazine length. Your .243 might not be long enough to hold and feed 6mm Remingtons.
What you could do is have a gunsmith cut about an inch or so off the chamber end of the barrel, and rechamber your barrel to .243 again. This should get rid of the freebore. A heck of a lot cheaper than a new barrel.