The symptoms of SI issues can vary wildly. You can have horse that just doesn’t want to engage their hind end, is “not quite right” behind, or you can have a horse that is a complete bronc at the canter. But all of those symptoms can indicate other problems too (sore hocks, ulcers, poor fitting saddle, etc.) Have you had a vet look at the horse and confirm that they think it’s an SI problem? If you haven’t done that, that’s where I would start. If you have some bony changes in the SI, your horse might benefit from an injection, but if you have ligament damage–an injection won’t help.
In my mares case, the ligament damage in the SI is being treated with stall rest, bute and hand walking. After that, we will start on a slow rehab program prescribed by my vet. I don’t think there is any plan to inject because there aren’t any arthritic changes or lesions.