Drop handlebars with drops that don't fit. Replace or adjust?
It looks like you're treating it as four positions: tops, hoods, vertical part of drops, horizontal part of drops.

Compared to my bike, your bars seem to be rotated forward, and the levers seem high up the bars. It strikes me that you might be able to try rotating your bars back and move the levers further forward. This would make the hood part of your bar flatter, in the photo when you're holding the hoods, the bar there seems to be going down-hill, and the hoods are going up-hill, but with by rotating the bar and moving the levers, that area would be flatter.

And, you'd end up with one 'in the drops' position where the bar would be angled at about 35/40 degrees, and you might well be able to both fit your hand there nicely and be able to reach the brakes from that one position.

I'm not sure if this Surly has the same handlebars, but whatever they are, they definitely seem to be rotated back more as I described. The ends of the bars seem to be vertically under the tops: http://lockouttoout.blogspot.com/2010/11/surly-long-haul-trucker-54cm-bike-black.html