First, how long was your horse on the track? If they were on the track for a longer period of time they are going to be more set in their ways as far as not cantering. I also wonder what kind of trot you have? Do you have the nice floaty slow trot, a medium trot and the racing trot?
Do you have a trail nearby? I have heard and found that sometimes the easiest way to get them to canter is to open them up on a straight away and ask them. Its not going to be comfortable but its going to help. Right now, it sounds like your horse does not know what you want of him/her. As a result, he is trying all the things that they use to want (speed and a pace). You need to let him know that a canter is what you want. A straight away on a trail would be one way to do that because its going to be harder for him to balance himself on the turns. In the arena, you are asking for leads and that could be confusing him. What you want is just the gait right now no matter how ugly it is. You can refine it once he realizes that that is what you want. So that's one thing to try.
I would in the ring focus on getting those three established and collected trots before asking for a canter. I only recently found one of my project standies wanted to canter. We have been working on doing that in a controlled manner. I am sort of in the same boat where we can get it going one direction and the other direction he just sort of falls apart. I am not sure what he does but is very smooth feeling, fast and not a canter.
I had one trainer teach a standie to canter by putting him on a small lunge line circle 10-15 feet and hit him with the lunge whip until he cantered. I would not recommend this. When I asked this horse to trot he would just canter. He was also a bit of a head case on the lunge line. You want to have smooth transitions between paces and not just one speed.