What a difference.
All I used was some Arm & Hammer baking soda toothpaste, a nice circular rubbing motion with my finger, and a soft microfiber towel to clean everything up.
Here are some before and after pictures, taken with identical settings in manual shooting mode (no editing, of course) -
So as you can see, it's a pretty huge difference. I'm not sure what kind of coating they put on the lens, but it horrifically degrades photo quality over time. I've had a case over the phone the whole time I've owned it, and the only glass/plastic that's scratched has been this back lens cover.
I saw another thread that suggested removing the lens - I don't think that's necessary. I'm back to stock-looking photos with just a little bit of polishing. Time will tell if it remains this way, though. If the coating was some sort of anti-scratch layer, it may get worse.