Glad you are having him checked! Better to be safe than sorry. Is your vet experienced with equine lameness?
I suspect my horse's issue has been going on much longer than I've known, as I've got practice videos where he swings his butt around instead of engaging it (but did NOT crossfire). Thought it was "just him". At 2:25 in this video from July is where I lope him through and we have our usual bad right turn. Back end looks fine there, as far as his footwork.
But I truly suspect he probably was starting to have a problem back then in July (or earlier), and it just didn't progress to obvious crossfiring until later, like here:
Let's just say I am going to be VERY suspicious for soreness for any horse that doesn't use his/her back end properly, from now on.
Your riding isn't really that bad (although hard to see because the video is small and far away). I guess the best way that I could describe it, is that it seems like some of the time you look like you are just out on a pleasure trail canter ..... not on the barrel pattern. I've taken English lessons so I can totally relate the opposite problem: I'm so used to keeping my weight to the outside, it felt so weird to put it on the inside for my English lessons.
One thing that you could do better is to LOOK with your whole body. When you go around a barrel, you seem to continue to keep your body orientated straight ahead. Instead, when you come around your turn, you should be turning your head and shoulder to look at your next barrel so you can head straight
Using myself as an example, as I come around the 3rd barrel, I'm looking toward home and starting to rotate my upper body in that direction too.
And I can't tell where you are looking in your video, but always make sure you do NOT look at the barrel. You don't want to run ON the barrel, do you? Nope!! You want to run around it. So look at the spots on the ground where you want your horse to travel. So especially to help him to not anticipate the turn, as your head to the barrel, pick your spot on the ground you want him to reach. And get him to that spot before letting him start to turn.
Every horse is going to work a little differently. For the horse pictured above, I would rotate my hips right before I was ready to turn. So if I was making a turn to the left, I'd rotate my hips counter-clockwise, and use my inside leg (back a bit) to kick during the turn. And I never really "sat" in the turn. I kept myself up. (Learned all this for him at a Lynn McKenzie clinic.)
My current horse Red? That way of riding does not work for him at all. And I'm still trying to figure him out (once I get him sound again, anyway), but I've got to make sure I sit and ask him to rate/collect before the turn and it's been working to put my outside leg back to drive him from behind.
So it'll take a little trial and error to see how your horse is going to work the best.