I agree with your assessment of it being the SI joint. I have had this problem for 9 years, which started on the right side. It started with the famous pop followed by pain. Anyway throughout the years I never stopped training or "working out", which I'm glad you've figured out you can and should continue to do.
As a pilot, I can tell you sitting for long periods of time made this worse. A lot worse. So from here on out during the normal course of your day don't sit for long periods of time if you're able to get up and move around. Just move.
Continue training but now more than ever it will be important that you really focus on engaging the abs and the trunk, which helps if you're really doing the valsalva as you should be. The belt of course also helps as a reminder to engage the abs.
I have had this issue so long that the pain moved over to my left side for several months and then just last week during REALLY light deadlift warm ups, my entire lumbar section/spinal erectors seized up. I could now walk upright, get out of bed without severe difficulty, get out of a chair, get dressed, etc. Never in my life have I experienced that and while I know plenty of people have, it wakes you up when it happens to you. I actually had to get an NSAID injected into me followed by steroidal injections the next day just to relax all that. Now I'm on borrowed time of steroids and need to figure this out. The thought is that for years my body had been overcompensating that right side injury that it finally wore out the left side and the ligaments doing all the overcompensating. I'm not a doctor but that's what mine told me.
I tell you my story to tell you that you're right on not just stopping the training with where you're at now but also keep doing the ibuprofen thing, use your abs, big air, and don't sit down for long periods of time. The ligaments and muscles get far too relaxed and otherwise disengaged, which in my experience makes it worse.
By the way I am embarrassed to admit my injury and famous "pop" happened on the leg curl machine that I had to use for a military strength test....