With the disclaimer that I am NOT a vet: From what you’ve described, I would assume SI is sore. If the suspensories are doing OK, I would start a gentle-on-the-body yet aggressive-in-focus strengthening program. Daily walking hills, backing up, then backing up hills. Stretches prescribed by a massage/chiro/vet, tailored to his personal issues - ‘waking up’ the muscles before work and teaching him that he can have this range of motion now. Long (15-30 minute) walk warm-ups over various types of ground - rocks, mud, grass, really anything you have access to. When he’s good and strong, not adducting at the trot (ie looks sound), you can progress to lunging long and low, naked or with side reins and an exercise band around the hindquarters to engage the core/haunches. If he loosens up with a canter, he can have a circle or two to get things moving - but no more. Start short - like 4 minutes of lunging and done short - and move up to 20 or so minutes over the course of a month or two, then adding canter a minute or so at time over the course of another month or two. Then rinse and repeat under saddle. If any backsliding occurs (starting to look lame/adducting), rest for a few days until he looks ok, then start back at a lower level he can handle and move back up. It’s long, frustratingly slow process but I’ve had 2 horses successfully come back from SI injuries with this method.
An SI injection may give him the comfort to be able to start moving correctly enough, but remember that it’s the strengthening that is necessary to fix things. SI injuries need a cradle of muscle etc. to stabilize them, and they will likely need that stabilization for the rest of their life - atrophied toplines on a horse who previously had an SI injury is a huge red flag that they may become sore again. He should also, if possible, be turned out 24/7; a large pasture with rolling hills is ideal.
The hooves will also need to be correct for this rehab to work - constant strain from incorrect angles will be difficult to combat and could lead to further tendon/SI issues.
I would NOT try to ride him through the leg swinging. I had a trainer who claimed my mom’s horse’s head bobbing/slight off-ness was a result of the way he was being ridden (‘rein lameness’). He was actually injured, and he was ridden in a way to mask the symptoms - eventually he was seriously lame, and lots of vet/diagnostics later found it was SI. Daily stretches, hill climbing, etc and he’s the soundest he’s ever been (knock on wood) and looks INCREDIBLE when he moves. There was immense pressure from the vet and barn owner to inject hocks. We didn’t, which was good because his hocks are fine. Gotta save those injections for when we need them 
(he’s 15 this year and shows no sign of needing anything). I personally would not inject hocks without a solid, hock-specific diagnosis.
There’s another horse at that barn who had similar ‘rein lameness’ that has been ridden/masked for years, and that horse is SORE. You touch her back and it collapses. She’s being held together with injections and previcoxx, which IMO is a terrible solution and unfair to her (I am not against injections in all circumstances, but I am against them being used to nurse a horse along and mask a problem so you can get one more show season out of them before dumping them quietly on the local classifieds).
I have a mare with chronic scratches (white legs and a barely-handled-not-trimmed-for-the-whole-summer-broodmare = years of untreated scratches when I bought her). I’ve been treating them, but given her history I’ve never been aggressive treating it, so I’m still squashing down the last bits of infection (it’s worse on her hind legs). She started recently having issues with holding her back feet for the farrier. She’s pretty stoic but if you pressed hard enough she was slightly sore over her SI, as well as tight through her hamstrings. I had the massage/chiro out; while she needed some adjustments she wasn’t ‘broken’ so didn’t need extensive rehab. She’s not perfect yet, but her hamstrings are soft, she’s no longer sore over her SI, and she’s accepting gentle stretching of her back legs. I would be surprised, if the scratches infection isn’t horribly irritated/weeping/bloody, that those would be the cause of the adducting.
I wish you luck in this rehab! It will be painful for now, but hopefully he will come back stronger and better than ever
He’s 5 and needs muscle - odds are these are baby things and he will grow out of it! I’m a worrywart too though so I get it. If anything I would back off the jumping and really focus on flatwork and polework (maybe some gymnastics thrown in for variety). You don’t want to create a rusher.