Well, with the 310 bore you can run anything. .308" bullets should run pretty good. I get decent results using 150gr .308" bullets out of my .312" M38 as long as they are NOT boat tailed. Flat base grouped much tighter than boat tailed when running in an oversized bore in my rifle.
You can run the .310" bullets, but selection is mainly in the 123-125gr range with a few heavier ones.
You can run .311" bullets at full strength as well. Pretty decent selection here.
.312" will work as well, but I see no reason to use them, if you do, just work up your load like any other reloading session. They will squeeze down without issue, I wish I could recall where I read a study about using oversized bullets, and the impact on pressure. It was an interesting read.
Me, I would start with 150gr Speer Hot Core SP, flat base, #2023 or any other quality flat base bullets. I like Varget or H4895 in my Mosins. I've run mine from mild to max, replicating military loads.
Then try the Speer or Sierra 150gr .311 offerings. See which gives you best accuracy. Have fun!
Note, .312 bullets ran 50-80fps faster than .308 bullets out of my M38.