Possible, but not nearly as inexpensive as buying a new barrel, or, in the case of the HandiRifle, a whole new gun to start with.
I priced out a rebore on a barrel, and it was about the same cost as a barrel from the same maker, and the grand total of a guarantee of success was effectively "cash up front, no guarantee".
A .243 is equal to a 6 mm. You will need to find someone that will ream out the barrel, accurately, and recut rifling appropriate for the 6.5x55 cartridge's bullet.
Any .257 bores available in the NEF? With a really fast twist to shoot the long bullets that are usually loaded? If so, they would be a better starting point.
Edit: My bad. Looks like you need a .264 barrel for a 6.5x55. A longshot possible would be to have a smith use the remains of the fitted barrel as a sleeve to sweat a Swede take-off barrel to make it mate up. Whether any of them WOULD, on the other hand, may be the obstacle.
Not telling you it can't be done, but really, how much are you willing to spend on something like this?
Cheers
Trev