I think we should keep the basic idea of having a marksman that is nimble and more in the firefight (the gunslinger) and another that is more static but packing a stronger punch (the sharpie). I just believe that we shouldn't go to either extremes however.
The main issue is that the SS is a (nearly) immobile perk in a very fast-paced game. What worked in the slower-paced KF1 doesn't nearly as well in KF2 (you could often find one of the best spot on the map and just sit there all game). Now, camping is still possible, but waaay harder. As such, the sharpshooter skills promote not only an unrewarding and unfun gameplay, but one that is rarely encouraged by the game itself.
I believe, as a result, that what should be done first would be to give him skills that doesn't affect his already mediocre mobility. Not giving him any mobility skills or passives is already kind of a disadvantage compared to the gunslinger (and other faster perks like the medic or zerk). Secondly, I would say that not crossperking any of their weapons would be a better way to differentiate them (and I'm saying so as someone who LOVES using the Centerfire as a gunslinger... and the .500 never leaves my loadout as a sharp). Bringing them other, unique weapons with similar roles would be the smartest way to prevent players from feeling nostalgic... and from both perks to step on each other too much.
Because yes, as Breadsticks mentioned : the guns themselves are great ! The LAR is still the king of the first tier, dealing very decent damages for its tier. The Railgun (and M99) is a killing machine for big targets. The M14 deals short work of anything, including big zeds if you manage to score a few headshots. It's definitely not the arsenal's problem.
I do have to nuance what Wytchphyre said, as I believe that scoring headshots is still the best way to reliably kill as a Sharpshooter. Less damaging weapons like the LAR and M14 truly needs that to deal with anything but trash zeds... Meanwhile, you'd be wise not to waste a costly shot of Railgun/M99, so aiming for the head is the best thing to do to quickly deal with an impending threat. I don't feel like "spray'n'pray" is a very valid strategy for the perk, unless you use the M14 against a swarm of trash zeds.
He however said it best : the sharpshooter is one of the two "nukers" from the game (along with the demo). The gunslinger is more in a harassing role. Oh sure, he can also dish out loads of damage very quickly, but you need a snappy aim ! Along with the zerk, he's excelling at "hit'n'run" strategies.
My last two points would be :
Sure, he's the "de facto" sniper of the roster. But I would say that it's mostly by default. He's actually more of a marksman really. A class that fires a few carefully landed shots, then move back to reload and bring the pain once more. He's definitely the perk that stands in the back the most... But that doesn't mean that he shouldn't move. He's like a "mobile" sniper rather than a "camping" sniper (enough references to FPS games, I promise)
The Mosin-Nagant, to me, is more or less an in-between gun. He has more raw power than the M14 but less ammo loaded and fires slower. Meanwhile, the Mosin is less damaging than the Railgun, but can load five rounds and fire slightly faster... It's really a nice middle-ground. But as the saying goes : "jack-of-all-trades, master of none". One would go towards the Railgun or the M14 depending on its playstyle, assuming he doesn't want to be a weird hybrid. And to me... That's definitely okay! Similar to how you can go for hitscan weaponry as a firebug or towards more "DPS" affairs. As a sharpie, you can either go for a middle-ranged sniper of some sort, that can adapt and retaliate more easily but dishing out less damage per shot (with weapons such as the Centerfire, M14, Mosin...) or on the flip side : a pure nuker of a class that will struggle in most situation but excel at one job : taking out important targets even before they become a problem.