16.3hh that's big for an x racehorse, do they have any comformational issues such as feet?
I'd be starting on the ground first, if they are unbalanced online they will be unbalanced when ridden. I'm guessing he/she lacks bend and looks out on the circle?
Is anything out, hips, bad back? Dumb thought but you are rising on the correct leg? Are they tracking up in walk?
You can teach sideways on the ground which lifts the stomach muscles gets them crossing, but this takes times and you only ask one step at a time and don't be too fussing on straightness early on.
If you lunge up to you, but i circle which is similar and i have a change of direction, the horse would do a turn on forehand or flying change if your cantering to double back. I only use halter or nothing.
Another thing i like to do is observe the horse freely, loosely in the school or i the field, the unbalanced trot you mention could actually be lameness?
Muscles take time to build up so i wouldn't do excessive work in the school i'd be out hacking building up tendons and hillwork.