Brifters pretty much require drop bars to be usable. If you were willing to try bar end shifters, there are a number of flat or flat-ish bars that would work well. (You would need bar end shifters, and new brake levers). Have you considered just raising the existing bars? You would need a new stem like the Nitto Technomic. They're available in a range of forward extensions, and would let you raise them about 3-4" taller than a stock stem.