As I found out the hard way (and with a bit of trial and error), there's a lot more to getting drop bars set up "right" than there is with flat bars.. the tilt/angle of the bars themselves, the shape of the bars, the drop, the reach, the position of the brake levers, the angle in/out of the brake levers (I like mine angled in *just* a bit), and it can get kind of complicated/finicky, but I think a majority of the problem you are having right now is really just due to the angle your brake levers are mounted. It might be "right" according to surly but my guess is that it's forcing your hands into a position that they probably don't want to be in. There may also be some other underlying fit issues that have you putting too much stress on your hands/wrists which may not help, but probably wouldn't cause a significant issue after only 2 miles - you'd probably notice more general fit related issues after at least 5-10 miles... so this seems like it may be different, and fairly localized. But at the same time each piece is part of a bigger puzzle and it's hard to solve just part of it alone... because other stuff can be contributing.
It may even be that a different shape or width bar is what you need... it can get kind of complicated because there are more variable involved. There really is no one size fits all with road bars... everyone is different and has different preferences. It's worth trying to find what will work best for you.
200 miles seems a long time to suffer before having a fit done, honestly, if there are fairly big issues that need to be fixed right now... if I were you I would take it back to the shop that built it up for you, and explain the issues you're having with your hands and the reach from the drops - I do think it's work taking the tape off and having the shop tweak them for you so that they're comfortable riding on the hoods *and* you can reach them from the drops. If you can't reach them from the drops right now that should be a big enough reason to have them moved now rather than later. You might even ride around without bar tape for a few days or a week to get a feel for it and make sure it's right for you (with gloves you might not even miss the tape much.. my mixte bars were "naked" for a week or so while I was deciding where my levers were supposed to be. You could even have cross levers mounted while you're doing this if you wanted.

