Some horses prefer a bit that conforms to their mouth more, like a double jointed, or shaped mouthpiece, including low port for tongue relief. They may also prefer a loose ring so that the bit can move up and down a bit in their mouth instead of being fixed - this also can reduce the effect of rider’s rein aids, especially unintentional (e.g. moving hands up and down while rising). Experiment, if you can.
Canter usually improves by doing more canter. Try going on a large circle, approx. 30 metres, and pick up canter, focusing on the shape of the circle. After doing one circle, come to trot and re-establish a good trot on that circle. Then canter again, on the circle. Repeat the up and down transitions, focusing on quality trot every time. As you start to be able to establish the trot within fewer strides, you can start doing your upwards transitions more frequently, until for instance, you can do 6 strides canter, 6 strides trot, 6 strides canter, 6 strides trot. You can also reduce the size of your circle to 20 metres. The transitions help to strengthen all the muscles needed to push-off and to go correctly (abs, topline).
A sensitive TB may start to anticipate or to rush, if you aren’t patient or clear.
Another exercise is to set 3 canter poles, 2 strides apart on a 20-30m circle shape. Work on your horse’s shape (curving through the centre of the poles), bend from your inside leg into your outside hand. You can still do your canter/trot transitions on this circle and around the poles if needed. Concentrate on the horse’s balance, the aim is to canter through the poles in an uphill balance with you in 2-point and a light contact on the mouth, and using your legs for impulsion and bend. Once you can do this, raise one or all of the poles, at one end only first, then you can do raised cavaletti. You can then also do the exercise of several poles on a straight line, keeping canter balanced, which is harder without a curve to help you.
All the best!