I tried completing Bleak Falls Barrow on Master difficulty as a level 1 pure thief, pure warrior and pure mage with no followers.
The 2-handed warrior had the most difficulty because he has no ranged attacks. I died and reloaded dozens of times. Died frequently with the others as well, but it was most challenging as a warrior.
Currently I have also invested a bit into 1-handed. I play most of the time with 2H weapons, but when fighting powerful spell casters like Dragon Priests, I started using the tactic of Slow Time, get a few 2-handed blows in, then switch to using shield with magic resist and a 1-handed sword. It's a bit easier to interrupt their spells by shield bashing spell casters.
Trying it at level one on
any difficulty is pretty insane if you're a pure 2H warrior, as the archers have elevation advantage and will shred you as you climb the stairs. I always wait until I have a decent weapon and armor, but then I don't mind heavily leveling Smithing early on via upgrading everything I loot.
I'm mapping out a 2H warrior right now, in fact, and she (going to be a female Argonian) will be something like this: Archery, 2H, Heavy Armor, Restoration, Alteration, Smithing, Enchanting, Alchemy.
I'd leave off the last two, but I play Dead-is-Dead and need the additional buffs. Restoration is semi-optional, but I have a feeling I'm going to need it, and Archery will be a secondary combat skill used when I can't reach, or do not wish to melee, an opponent.