There is an easier way. The 6mm Remington is a dandy cartridge which happens to be the 7x57 necked down. Why not have your double rifle in 6x57R, being a 6mm Rem loaded on the 7x57R case?

You could even save yourself some money by forming your brass from .444 Marlin, which is really cheap.

Back in my younger years when I was having a lot of custom work done I had a Win High Wall made up as a varmint rifle in .257 Roberts Rimmed, and I made my brass from .444 Marlin. I also made 7x57R brass from the .444 when I had a double rifle in that caliber many years ago.

Likewise, you can form .243 Win Rimmed from .444, but why do that when you can do the 6mm Rem (6x57R) easier and without the big trim job.

Just my $0.02 worth.
Curl