I certainly agree that for accuracy the bullets must fit the bore but they also must fit the chamber throat, that happens only with the full length Nagant case. Nagant chamber throats measure .337-.339" to make room for the full length case. If accuracy is your goal then using a case which is .030" smaller than the chamber at the base to fire a .314" bullet through a .338" chamber throat to then slam into a bore with no forcing cone does not seem like a good recipe.
I load Hornady 90 grain bullets of either SWC or HBWC form, both of which are .314" diameter. To load them into Nagant brass I expand the case mouth with a Lyman "M" die intended for loading cast bullets in .30 caliber. It is a two step expander, the second step of which expands the case just large enough to start a .314" bullet without shaving lead.
Old story "if I'd known then what I know now" I never would have wasted money on the Lee die set. The only piece of that I actually use is the seater and that had to be modified. The sizer overworks the brass, reducing it much smaller than need be. A .30 carbine die is better for full length resizing, reducing the case just enough to slip easily into the chamber and I could have modified the .30 carbine seater the same as I did the 32/20 seater which came with the Lee die set. The Lyman "M" die must be obtained separately, likewise the RCBS shell holder, so the Lee so-called 7.62 Nagant die set is really pretty useless.
The factory loads with undersized bullets really don't shoot all that badly. The best group I've gotten from my Nagant was with the Russian Match ammo, 1.5" at 25 yards, but it is Berdan primed. Fiocchi Factory loads went 1.9" and PRVI went 2.8". My best handload with the 90 grain Hornady SWC over 3.5 grains of Trail Boss grouped into 1.7". The 90 grain HBWC did not so well, 3.5 gr. Trail Boss went into 2.6" and 3.2 grains HP-38 in 2.3", all groups at 25 yards rest.
The photo shows the bullets, the Lee seater with the modified seating stem, the Lyman "M" die and a .314" Hornady SWC just started in the case mouth and ready to go to the seater. It all fits.