Your problem is the plastic frame holding the window in place. You could probably make one out of plywood, mdf, pine or whatever. Cut the shape, router the edge to get the curvature their now. Prime it, paint it, install it and touch it up. It shouldn't warp like the plastic does. A cabinet shop shouldn't charge you too much to do this if you do not have the tools. Cheaper than replacing the door and it sounds like there is nothing with it, just the window frame right?
Check the door manufacturer for a new frame. What color is the door?? Light colors do better than dark. But the storm door is the culprit. Modern storm doors really work well, keeping air trapped, and keeping heat in. Use the screen in warmer weather, glass in cooler weather.