The mustache bars will stretch you out a bit more if you ride on the curves. They also tend to run a little wide. So, if you're riding in close traffic, they might affect your ability to maneuver effectively. I like them, and think they offer many useful positions, but I prefer drops.

I also ride bullhorns. They offer fewer hand positions. They also stretch you out if you ride on the horns. They're not as useful as drops when you're climbing, in my opinion. If you ride with the bars at saddle height, they will not really be useful. You could just add bar ends to your straight bars and get essentially the same effect.

If the drops hurt you, and you cannot ride with the bars below the saddle height, you might consider that the frame may be a little too small for you. Unless there's a medical/age reason, you should be able to ride with the bars slightly below the saddle. Your back should not curve 'too far', as you put it, if the bike fits you, although some curvature of the back is expected. You can get drops with more 'spread' in the drops, this should make it more comfortable.