I asked the barn manager to film me jogging Tris out on Monday afternoon so I could send it to the vet. Good news: he actually looks not half bad! He's not 100% sound (that's too much to hope for) but he looks good enough that the vet was impressed. I've asked her if he looks good enough to start handwalking; still waiting on that answer.
Walk and trot from behind:
And walk and trot from the side: