I would definately work on getting him very soft and responsive with the slowing down and stopping firstly from the walk, then from a trot, when you can do it at the trot, keeping a nice steady pace without having to keep constant pressure on his reins or having to heave to pull him up then start at the canter, but only canter a short distance at first, like just 5 or 6 paces then down to the trot when he does that calmly then up the distance, but remember if you can't get him to be soft stopping or slowing at a walk trot you won't be able to at the canter, he is possibly just really excited to be cantering and not happy about stopping, also some horses like my arab, you have to be a very polite rider, what i mean by this is a small warning of what you want before you want it,
imagine if you were enjoying playing or watching a movie, and mum suddenly bellows out to do the dishes or else, you would be quite annoyed right? but if she asks you to do the dishes soon please and then asks again a few minutes later you wouldn't be as upset with her, when going down in transitions like from a canter to a trot sit deeper and back a bit, start saying whoa or slow or trot whatever voice aids you use as the warning, then use gentle pulsing pulls with the reins not hard yanks, just gentle pulls with a release he won't be " cranky" because of a hurting mouth, a lot of the time being more gentle gets better results as a lot of horses will play up with head shaking or cloud sniffing ( head straight up) and they are not trying to be bad, they are just trying to get you to realise OUCH ! you are hurting them and they are happier and more responsive with less pain, i had to change to a softer more gentle bit with 2 joins in the middle so my thoroughbred wasn't getting dug into the top of her mouth like with the single join snaffle she really hated that one and was much better with the softer bit, easier to stop, and turn etc.
when you ride with a tight rein you actually push all the energy of the horse into the hind quarters and they can feel like they are waiting to explode forward i would try riding him in a small area with a loose rein with very gentle hands and gentle commands at a walk first so he can get used to you not holding him back, then at the trot with loose rein, by loose i don't mean hanging down on the ground loose exactly although i can and do ride my arab with reins that hang down in a big loop, just with basically no pressure on his mouth, he just may be trying to out run the pressure if you can understand what i mean? i definately wouldn't practise this outside an enclosed area at first as you may be nervous and tight yourself which will just tell the horse there is something to be worried about so he will be edgy waiting for the "thing" it is amazing how much the mood of the rider can affect the horse, but think of it this way, he is trusting you to keep him safe, so he is going to reflect what ever energy you are giving off, if you are relaxed and calm he most likely will be too but if you are sitting tight on his back, possibly gripping with your legs ( which is just telling him to go faster) but at the same time pulling with the reins, telling him to go slower that is confusing to him, only do what you feel safe and comfy doing you will get a much better ride that way. sorry this is so long but i feel very strongly about being gentle with horses lol a perfect example is a friend of mine she is only 17 and has done pony club for 3 years on the same pony, she was always holding her back hard and tight with the reins especially at the trot and canter, having the mare jig jogging ( prancing sorta) always throwing the head and basically didn't have much control of her mare, she even used a hard dutch gag bit thinking she had to get harder to get more control.
i got her during one ride to hop on my nice calm sleep walker mare and made her take note of how relaxed she was and how very loose the reins were they were hanging way down on her neck, then told her to get back on her mare and to stay exactly the same, loose rein and very relaxed instead of tight and waiting for the explosion, well you should have seen the difference in both rider and mare, brought tears to my eyes, her mare didn't prance or try to bolt or shake her head once, it was like she was saying thank you this is how i want you to ride me, her head was nice and low and relaxed instead of high and tight. we rode about 40 kms that day, the first 6 were hell for her, then with the different riding style we showed her the rest were so very relaxed and calm, even with big cattle truck going past i just told her to trust her mare not to play up,( we already knew the mare was fine with trucks ) to stay calm loose rein and yep her mare was just fine. we didn't try trotting that day as i wanted them both to get used to the new riding style but she did trot the following day with loose reins, then even surprised me by actually letting go of the reins totally riding with no hands, ! it was beautiful to see. basically horses are happier and more willing to do as we ask the more gentle we are and the less we hurt them, just makes sense really.
best of luck, as for the change in bit issues, maybe your mum can buy you a softer different bit and or bridle for you to use with him, or maybe you can borrow one to try him with, then if he goes well show the other person or owner the difference and they would most likely want to change to what is working better.