I experience alternating pain, too. This happens with my hips as well as my SI joints. For the first few months of having AS, I had pain exclusively in my left SI joint. Then, one day, I suddenly felt it on the right side. Now, it switches back and forth randomly - sometimes both at the same time!
The only thing I can recommend for hip pain are specific stretches and topical creams like BioFreeze or Voltaren. The cooling sensation of BioFreeze helps my hip pain. However, I know that's not a solution.
Have you and your rheumatologist considered other medication options?