Just shoot the one you can shoot best.
Any animal you shoot won't be able to tell the difference what it has been hit with, if you do your job and put a good bullet into the boiler room (lungs).
If you are one of those that insists on breaking the shoulder knuckle, use a tougher bullet.
Personally, I prefer a bullet that goes all the way through and expands well, while retaining at least half of its original weight.
The difference between the two cartridges you have is mostly technical but not different enough to make a real difference in how well the animal goes down.
If the barrels are heavy weights, you may want to turn one down, so that it's comfortable to carry and if necessary, shoot offhand.
The rifle that needs the ways opened to feed would be the latter choice for me.
I've used 6.5 bullets to take Elk, Moose, Bears, Deer, Wolves, Coyotes, Marmots and Gophers. If you use well constructed, 130-140 grain bullets, hit the vitals well, there won't be any issues.
To give and example, a now deceased but good friend and I went on a Moose hunt to Pink Mountain in northern BC, back in 1999. It was a late season hunt, early November. Cold, wind, driving snow, you name it it happened. Cleared up and the temp dropped to -25 during the day.
There were 6 of us in all on that hunt, as we met some fellows up there that had LEH draws as well. To make a long story short, we all ended up in our tent, because it was the only one with a decent stove.
After the 2nd day of the hunt, the only rifle that was functioning properly, was my Remington 700 in 257 Roberts.
All of the 338 Win Mags, 300 Win Mags and a 30-06, that all there considered to be the absolute lightest cartridges and calibres for moose, were non operational.
The first day, one moose was shot with a 300 Win Mag, half a dozen times. Both front quarters were blood shot.
The rest of the moose we took, 5 more, were all shot with my 257 Roberts, loaded with 120 grain Hornady Interlock, flat base, spire points, over a crushed load of H4831, over CCI 250 Magnum primers.
Every one of those moose went down within 20 yards of being shot. None required a second shot.
One received a second shot in the head because the hunter I loaned my rifle to, just couldn't believe the animal was dead, or close to it.
One thing, all of these fellows, were being extra careful about shot placement. In their minds, the extra power their personal rifles had, made up for the lack of energy delivered to the animal.
Now, at longer ranges, say 300 meters or more, that extra horsepower may be needed. Then again, if the shooter is doing their part, maybe not.
I have always been impressed by how much a small calibre bullet, driven at moderate velocities can do and underimpressed at the big, hard kicking magnums.
I used to use a 338-06 for everything. I got the big boomer bug. I'm recoil shy though and hated shooting the 338 Win Mag with a passion. So I went to the 338-06 with 225 grain Hornady Interlock, flat base, spire points at just over 2500 fps.
Then, a worn out rotator cup even made that to painful and for the last five years, I use the 6.5x55 for everything.
Your .260x308, will do anything the 6.55x55 will do.
Your choice. Your labor and fooling around with the rifle. As long as the extractor will work on the 308 case rim, the 260 would be my choice