You could build up a Calfee Luna Pro easily in that price range with a comfortable riding wheel and your pick of parts. The Luna is a awesome bike for the money. I have one along with a Dragonfly. One idea... I prefer building up a frame since you can pick exactly what you want...you can save money getting an off the shelf bike. The advantage also for you is that if you go somewhere with someone who is great at fitting you hopefully the components can be set up to help your back.
The problem for me is the bike isn't why I get dropped on hills...it is me...