So I just thought I'd give an update on this.
After my instructor finally fought him through it, I tried a few more times to trot him.
Mostly my method was to ask for the trot. When he didn't give it I'd push him around on the lead rope with the dressage whip and get him into a trot (like mini-lunging), then when he'd made a full circle back towards the direction we were initially heading I'd trot off alongside him. I also initially did this heading back out to pasture, so it was somewhere he wanted to go already.
As soon as I got just one step that way I'd start petting him, telling him good boy, and giving him some carrots. Then a couple steps.
Now here we are a couple weeks later. I just say the word "trot" and he immediately picks it up, no matter where we are and where we're heading.
We're still working on speed control - he tends to go faster than me, but not significantly faster so I can work with it.
So great progress. I just had to figure out how to make him understand what I wanted.