Sharpshooter is an incredible trait once you get it- Even your basic pistol does 33 bullet damage per second (with the starting T-mod applied to it). So you need to survive that long.
First, trait order SoG3 -> EE3 -> MSs is fine in my experience. You may choose EE1 instead of SoG3 at level 3, but IMHO the firing speed is better. You should start with a T-mod - apply it to your starting pistol on turn 1.
The key to survive early is to play cowardly. Like, really cowardly. Find a sweet spot so that enemies may only come to you from where you can easily cornershoot them. With SoG3 you do enough damage that everything up to pinky demons have little chance getting in a shot at all. Don't explore the level - everything will sooner or later come to you. Setting the Run-Wait command to some very high turncount in the config file helps.
Also, check the number of enemies on the floor (via @ key) each time you enter a level and every now and then. It helps for keeping track.
If you are not playing AoMr, always carry a shotgun and a few spare shells for crowd control (especially for tough starting positions).
Instead of lowering your difficulty level, consider raising it to UV. You'll have even more deaths, more enemies, and hell knights in the fray at level 3, sure - but the increase in XP is enough to get you to XL7 before Phobos Anomaly, where Sharpshooter is tremendous.
This connects to my suggested special levels:
Skip Arena, unless you care for the badge. Once you have the badge, Arena seems pointless to me. Another option is to clear only the first two waves and then leave, to have normal Chained Court.
If you skipped the Arena, Chained Court with five Berserk packs should be a breeze - two zerk for one half, two zerk for left half, destroy the two boss monsters and that leaves you with one berserk to heal yourself before exiting. The two mods are usually well worth it.
If you get Military Base, you're lucky. This level is fairly easy for a pistol build, just stand at the start and cornershoot everything. Once they stop coming, clear the right half (careful about those three elites in the bottom right corner, though!) then the left half. On UV, this is approximately where MSs kicks in, although at least once I even reached level 8 there.
At Phobos Anomaly, use a phase device when theambush walls open. You'll almost always land in a better position. Alternatively, blast the right door open before triggering the ambush and rocketjump through - two rocketjumps should do the trick for little damage. You'll be in much better position to handle the initial wave.
Further on, it's only about how you feel. Even on HMP, MSs should kick in at the Armory/Deimos Lab, so your damage output should be high enough to offset shamblers' regeneration with little trouble.
Worthwhile assemblies are everything that boosts your speed even more. Tactical armor and boots are particularly helpful in the early game.
Energy pistol is great against revenants, ignoring their bullet resistance (the same goes for lost souls). In particular, it will allow you to do The Wall/Containment Area, especially if you already have MSs. Also, City of Skulls/Abyssal Plains are doable with an energy pistol (it's suicide with bullets only).
The first P mod you'll eventually find means that even a normal P-modded pistol with MSs does 13 damage, which means it does reliable knockback.
But what you really want is to find a combat pistol and some P-mods. This is an outright godsend - with three P mods, MSs and SoG5 you do incredible 230 damage per second, and with further two B mods you have enough ammo capacity to take down Cybie or Spidey before they can even fire a second shot.
There. Hope it helps a bit :).
EDIT: Oh. Ninja'd.
