I've been looking at it from the perspective of an alpha strike - how do you get the insanely good alpha strike as early as possible? From that perspective, I think Gloomstalker 5, then Battlemaster 4, then Assassin 4, then Sorcerer (many options for which type) 7. Human Variant makes sense for that early feat...
So, Human w/ Sharpshooter, Gloomstalker to 3, using a longbow. Assuming an enemy AC around 14:
1st round: Bonus action Hunter's Mark, 2x attacks for a total of (2d6+3d8+26) * (0.45 to hit) = 20.9 avg dmg
Later rounds: 1x attack at (1d8 + 1d6 +13) * 0.45 = 9.5 avg dmg
Compared to Crossbow Expert, you give up a little damage in the 1st round, but get more back in following rounds:
1st: BA HM, 2x attacks for (4d6+1d8+6) * 0.70 = 17.1 avg dmg
2nd+: 2x attacks for (4d6+6) * 0.70 = 14 avg dmg
If you think the combat is going to last two rounds, you're better off with sharpshooter since you don't get the redundancy of bonus action in the 1st round.
Compare to Battlemaster 3 with Sharpshooter and Precision attack, using a longbow and action surge in round 1:
1st round: 2x attacks at (1d8+13) each, with a base to-hit chance of 0.45, but boosted to 0.675 with the help of precision attack: (2d8+26) * 0.675 = 18.2 damage
2nd+: 1x attack at (1d8+13) with 0.675 to hit: 11.8 dmg
So your instinct is right that starting gloomstalker puts you a bit ahead when you first get your subclass.
How about at level 5? I'll just show the 1st round damage, since you're prioritizing the alpha strike:
Gloomstalker Sharpshooter + ASI to DEX, Hunter's Mark in 1st round:
1st: (4d8+3d6+42) * 0.50 = 35.3
Battlemaster Sharpshooter + Crossbow Expert, Action Surge, Precision Attack:
1st: (5d6+65) * 0.675 = 55.7*
*strictly speaking it will be a little bit different from this, since there's a chance you'll blow through all your superiority dice in the first four attacks. But it's not a big chance, and it's also partially offset by you never missing by more than 8 in those first four rolls.
So the Battlemaster can blow the Gloomstalker out of the water in the first round once per short rest at level 5; whereas the Gloomstalker will be ahead the rest of the time. Pick your definition of alpha.