As I understand it there were no special production Mosin Sniper rifles. They simply selected the most accurate ones when testing new production rifles and used them as sniper rifles. The Russians don't really use true snipers. They are really designated marksmen and very widespread in their use. The sniper mosin was really only good out to about 300-500m which was fine for Russian military doctrine. Beyond this it typically wasn't accurate enough and the scopes of those times were pretty crude. Even the Enfield sniper rifles which were actually specially manufactured to higher standards were not really good for much beyond 500m. Similarly for the American Springfields, so that mile crack in Saving Private Ryan is complete BS. Don't expect anything like what you can get out of 6.5 Creedmoor for example. If you want the true history of a genuine WW2 Mosin Sniper good luck they are very rare. Almost all are actually reassembled from old parts but are not truly a matching sniper. If you just want to shoot one in that style then a repro should be fine provided the rifle is decently accurate, a big question with any Mosin. They were typically "used hard and put away wet". Mine is about 6MOA on a good day.
What you describe as "no special production" Mosins is accurate only for Izhevsk, it was different for Tula.
In Izhevsk, they had no special sniper-specific production lines, instead, they test fired every single barrel, and selected only the best barrels to a regulation "sniper standard", to be fitted on special highwall receivers, which would themselves be fitted out with their optics, and be sent to the further steps of production and issuing.
In Tula, unlike Izhevsk, they had dedicated production lines for sniper rifles, where barrels would be sniper-purpose made from the start.
Real sniper production Mosins, in order to be issued as a sniper rifle (instead of having being tossed in a pile of regular infantry rifles), had to reach a standard of 10 shots into a 3.5 cm grouping at 100m, 7.5 cm grouping at 200m, 18 cm grouping at 400m, and a 35 cm grouping at 600m. A standard which is easily surpassed by modern standards of high quality commercial rifles, however is a rather good factory precision standard in its own right given the context of its production and peers, and considering that it is far better than the average of the "minute of man" precision of pretty much all contemporary regular infantry issue rifles.
Also what you say about the Soviets having a different sniper doctrine is very true, being more close to contemporary support Designated Marksmen roles.
I haven't properly shot and measured my (non-reproduction) PU Mosin, but from very casual shooting, it has gotten a solid 3 MOA, though the barrel on mine looks barely used and well preserved.