I don't believe there is any inherent accuracy edge of one cartridge over the other. Generally speaking, to get the same Ballistic Coefficient in the .308 bullet you have to go about 40 grains heavier than the .277 bullet (180 gr to 140 gr), which would have more recoil. Less recoil is always conducive to more accurate shooting.
If both rifles have the same barrel length, you should also achieve slightly better drop numbers with the .270. Using the same Hornady SST bullet in both cartridges with max published velocities and a +/- 3" point blank range sighting, I came up with the following drop numbers:
180 SST @ 2850 fps = on @ 241 yds, PBR of 283 yds, -68" @ 600 yds, -319" @ 1,000 yds.
140 SST @ 2950 fps = on @ 249 yds, PBR of 293 yds, -61" @ 600 yds, -285" @ 1,000 yds.
These numbers are close enough that if one rifle had a longer barrel to achieve greater velocity, or if one was consistently more accurate, I would consider that a greater advantage. You can compensate for drop, you can't for innaccuracy.