So if it turns out to be .308 what is the implication? No military surplus ammo for safety reasons? Or no military surplus ammo for barrel life reasons? Or perfectly fine?
I guess it depends how anal your impression of safety is. Please don't take that personally.
Your rifle and many more like it were made to shoot the issue ammo of their respective nations safely. The chances of a kaboom or any other damage is so minimal as to be considered not there.
a couple of thousandths of an inch will slightly raise pressures but they will still be within operating parameters. The worst effect will likely be noticed in accuracy and even that will likely be minimal.
The steel cores of the ammo in question is relatively soft. Not as soft as lead though. The jacket around the core has enough room for obduration to swage down to fit the bore as it enters the leade.
IMHO go ahead and shoot whatever you find. I would be more worried about unidentified hand loads than surplus or factory offerings.
Your rifle is old and if in decent condition should be strong. I also have an all matching M91 with SA marks and a .3085 diameter bore. It's a 1915 New England Westinghouse made in the US. I have no idea whether this rifle went to the Russians and the Finns later captured it or if it was sold directly to the Finns out of the US after the Soviets took over and the order was cancelled. I suspect the latter. The bore rates 8 out of 10 conservatively. Condition is VG+ all over.
It could care less whether the bullets are .311 or .308 diameter. It shoots them all into a group between 1 and 2 inches depending on whose eyes are behind the sights.
My go to handload which I use in all my M91 and M91/30 rifles is is 50 grains of IMR4350 over cci 250 primers using Sierra 174gr, .311 fmj match bullets which have lead cores. I also have a bulk pack of appx 180grn, .311 diameter pulled bullets I purchased from International around 30 years ago that has cupronickel jackets over steel cores. I've used most of these up now but I had 5000 of them at one time. I use the same load for those as the Sierra offerings. There really isn't much difference in accuracy.
This thread got me thinking again so I looked at my personal loading manual which is often firearm specific.
I use the same load for 180 grain FBSP bullets as well. I don't use these anymore as they are intended for hunting. They are old and made by CIL and have exposed lead bases, which likely help them to obdurate to the bore quite well.
Just as an example, I also have one of the first No4 MkII rifles built by Pakistan Ordinance Factory. It has a .3095 bore. It doesn't seem to care whether it shoots anything from .308 to .312. IMHO, the Mosin design is every bit as strong as the No4, maybe stronger, although they are originally intended for cartridges loaded to similar pressures.
About the only difference you might notice between cartridges, surplus or commercial is how fast your barrel heats up.
Shoot that old girl, keep her clean and she will outlast you and maybe your children, barring political ineptitude.