If you dig, you find that alot of military contracts were "let" where things were not exactly as expected, but "met" contract standards. I think alot of the first 44's/M-Nagants were made in USA by Rem and Win. While, I suspect, the ammo called for a .310 bullet, rather than set up to make a different barrel, Win & Rem just chambered .308 barrels ... 44? From Finland? Captured Russian that got "reworked" by the Finns? Or just captured? Or made by Finns? Or...
Yes, slug the barrel. You can use a buckshot (0 or 00) maybe or a muzzle loader or C&B ball. Come in from both directions with wood dowels and tap smartly on one to upset the ball into the rifling... Lube first with a squirt of WD 40 or oil or about anything. Slide the ball out and "mike."
Or cast it in sulfur. You can get "garden sulfur" at better gardening departments. (I think it is for making soil more acid.) Melt it. Outdoors or in the garage. Camp stove works fine. It is not explosive, but if you get it too hot and it catches fire, IT STINKS. Mama might expell you from the house for a long time... Pour thur a funnel into the barrel. Or go in from the back and do the chamber and throat too. Stop the barrel first with a cleaning rod and tight patch. Lube a bit. WD 40 or about anything else. GENTLY push cooled cast out. If it breaks, start over. Any machinist can measure this and give you guidance.
From there, you start. Go buy the bullets that are closest to filling bore and groove of your gun. (I measured a couple P14's with a friend and got Winchester barrels for .303 at .307 groove...
Been planning to try .308 bullets but haven't gotten it done yet... ) Shoot them and see what happens. If that is not good enough, then you might want to try the next larger bullet. Less often, the smaller bullet might work better. Basically, you need bullets that your dies will work with. If you have an expander ball for .311 bullets and you try .310 bullets they may not be held in the case neck. Slide them in and out with your fingers. .312 should work o.k. And if you shoot best with .310's, then you contact the die maker for a smaller expander ball unless you are handy with tools and want to sand the one you have down with emory cloth or sandpaper or??? Chuck it in a drill and spin it and apply the polishing compound...
As pointedly said, this could be the diffrence between a shooter and a tent stake. LUCK. HAPPY HOLIDAYS.
