"Can I use .310" bullets in a .308" bore? "
It's highly unlikely to cause any problem. Or Barnses wouldn't make them, surely knowing they will be used in .308 bores! (Few of us load for the .310 Martini Cadet anymore!) Being light means it's short so it will get a good running start before hitting the lands and then, after traveling its own length, it will be bore size.
The only potential for a problem would be IF the loaded round is large enough to pinch the bullet in the neck and not release it when fired. Since most factory chambers are plenty large and most case necks are fairly thin, that's not likely to happen. Only way to know is to check it. Load a dummy round, fully "paint" the neck with a felt tip marker and then chamber it. Pull it out and check the neck paint rub. Unles you see a complete circular wipe off around the neck there is some clearance you will be okay. If there IS a full circle wipe, don't do it.