I busted my SI joint pretty good a couple or so years ago pushing sumo deadlifts too hard without regard to my form and it'll still flare up on occasion. I did a lot to heal it. Glucosamine and fish oil, the hip ab/ad-duction machine with a super slow tempo and pauses at peak stretch and contraction, cross frictioning, lax ball and foam rolling work, different types of specialized mobilizations stretches and activation work. Basically, I used the throw a bunch of shit on wall and see what sticks approach lol.
It was about a month before getting in and out of a car stopped making me wince. It was 4-5 months before my squat got back to where it was. It was 9-10 months before my deadlift returned to where it was.
One of the most recent changes and most helpful in preventing flare ups recently is making drastic changes in my squat and deadlift form. Basically, I used to do everything I could to take advantage of my hips and a reduced ROM. I sumo DL'd very wide and squatted very wide sitting back a lot low bar.
Now I do the opposite. My squat is just about an olympic squat cut high by oly standards with a high bar position. My deadlift is now a very close stance conventional. Granted it took a while to get to my previous numbers with the new form, but once I did I could keep progressing without my body just quitting a lift on me because of my hip. I can squat 35 more lbs with the new form and deadlift 40 more lbs while cutting weight from 200-180. I don't think that's too bad for just over a year considering the weight loss and how drastic a form change that was.
You might also want to take a video of your squat from in front or behind to see if you're unconsciously pushing harder with one leg than the other. If you are, basically focus on not doing that anymore and waking up the sleepy side. Activation drills and one leg work can help that. But if the issue is really SI related and you do single leg work, make very sure that your hips are level and not leaning to one side or you'll just aggravate the issue.
I've heard of SI belts. I thought about getting one but after the form changes decided I didn't need it. Just something to look into.
I don't know if any of that helps or not, but I hope it does.