I'll chime in with my $.02..............

Try the drop bar (or moustache), you might really like it. It you've got the stem length right and bar height set up close to saddle height, there's no reason it won't work well, current fashion be damned. Use a shorter and higher stem so the brake hoods on the drops will be right about where the ends of your flat bar are now.

One thing to look at when climbing is your hand rotation to use your biceps effectively. Having your hands on a set or brake hoods with drops, or on bar-extensions with a flat bar, puts your hands and arms in essentially the same position.

Drops with bar end shifters give you multiple hand positions, good climbing leverage, and (dare I say it?) a friction shifting option.