Be very careful with an inversion table as inversion therapy can elevate blood pressure to dangerous levels. If you suffer from hypertension at all you should probably avoid inversion therapy.

The SI joint is really deep and hard to heal. Ice therapy usually won't get into the area because it's too deep, but you can try icing the area for 20 minutes on, then at least 40 minutes off. Take some anti-inflammatory and do light stretches. If the doc thinks it's from a muscle imbalance, then you need to exercise ALL muscles in the area, but add a little more emphasis on the glutes.

Keep an eye on this and see if the pain migrates at all as it could be other than an SI joint. It could be a disk problem or sciatica or something similar to that.

Just take it easy and do all your therapy and treatments. If it hurts, then don't do it, but you have to differentiate between pain and irritation from the activities. Not always easy.