I am going to second Tiffani B and others who have said to back up, FIX the issue without a fight, before asking horse to canter again in the cart.
Has this horse EVER been worked in the long lines, pulling a tire, and asked to canter? Our horses being trained to drive ALL get worked at Walk, Trot, Canter on the lines, wearing harness, BEFORE being asked to pull the tire. Then they learn the feel of tire pulling at walk and trot, UNDER control, voice commands being followed BEFORE getting asked to canter pulling the tire. Tire is off a pickup truck for our larger horses. Smaller animal needs a smaller size tire. It is NOT A WEIGHT pulling training step. Tire is just for dragging resistance, is moved EASILY by the horse, adding much feel to the harness as tire swings out, moves around, while being pulled along.
Horse MUST be firmly under control of handler on the long lines, so any odd reactions, spooks, are able to be corrected with a halt, calming words and lines to the bit with the word WHOA resulting in a STOP by the horse. If he will not Whoa as asked, with no tire at Walk, then Trot, you need more training time, before adding the dragging tire behind. This Whoa thing is REALLY important in a Driving horse, it is the Emergency Brakes!! Probably all the Drivers on here have NEEDED that dead-stop Whoa, using only their Voice, at least one time in their Driving experiences!! Horse MUST stop dead, RELIABLY, for you with the Voice, to move on to other training steps in Driving. Could be your āSave the dayā moment in the future!
Letting out the backstrap, giving crupper more length for horse to lift his back in work, is an excellent suggestion. Remove possible problems, so horse has no excuses to react badly.
Sorry, with your CTJ sessions, you are just creating problems for the future. Probably going to hurt yourself and NOT getting the horse to āfigure outā what you want, by making him buck. Part of an old saying is āThere isnāt a Cowby that canāt be throwedā, and it SOUNDS like your horse is getting better at removing you than he started out being. Do NOT want him learning that!! You are not going to āfixā his bucking by using a bronc halter and rope, for bucking while being ridden or driven.
You need to back up, check as suggested for possible pain issues first. Then if none are discovered, adjust the harness fit, saddle fit, to remove excuses of surprise pulling on tails or pinching, when asked to do something you want. Then you TAKE THE TIME to do a good warmup, get his muscles loosened, warm, so he does not get HURT with sudden start off cantering, or changes of pace. It is way too easy to damage the legs with starting work cold legged, cold backed. NO winning athlete, good horsemen in Competition, starts their work cold. They ALL do a Warm-up routine to soften muscles, get the blood flowing and PREVENT problems from happening. Another old saying, āWalk out the first mile, and home the last mileā is part of warming up the horse before using him. Less likely to hurt him going out, and gives a good cool down coming home. We do a 20 minute warm-up here, 5 minutes of walk, 5 minutes of trot going one way of the arena, then the same going the other direction of the arena. Horse is nicely warmed up, moving smoothly, ready to go to work now. Warm up session is for EACH time we use them, every hitching or training session on the lines. Equals the time of the mile walk going out to drive down the road, in warming up the horse.
Still wondering, if your horse knows how to work on long lines? They are a wonderful training tool, let you do many things with horse, while keeping you safe from harm as he learns.
Guess I would make that kicking strap part of his permanent harness equipment for a year or two. Better safe than sorry.