I realised that my first post was somewhat vague so I thought I would add to it.
First of all, being an alpha horse in the barn doesn't translate to being a dominant horse under saddle. The only reason I am mentioning this to you is because if you have in your mind that he is an alpha horse and that is why his behaviour is so bad, you have JUST made the first 'excuse' for his behaviour. So firstly, get that thought safely out of your mind.
Now, if you are going to ask for a canter and he bucks, he still ABSOLUTELY must canter. If you stop him and go back to a trot straight away - he has won that round. If you don't feel he is ready to canter, don't ask for it. Trot him until you feel he is responsive enough and/or worn out enough.
What I would do (and believe me I have done this with many a TB off the track) is get out of the arena for a bit and find somewhere with a bit of space - trails or a large pasture/field. To do this with a bucking horse will require you to be able to sit quite a few bucks and not be bothered by it so totally up to you as to wheter you want to try it or not. I would take a crop and use the saddle that best fits your horse so that you can eliminate another 'excuse' for him.
Now for the hard part- Once he is warmed up, ask for that canter transition. If he bucks he gets a good smack on the bum but he MUST canter. Once he is cantering smoothly - keep going! This can take some time/space hence why I like to do it out of the arena. Sometimes for a young horse, particularly if they are off the track, it is hard to get a nice smooth canter in only 60m but if you give them adequate space they will sort themselves out.
Once he is cantering nicely for more than 20 strides, tell him he is a good boy, move down to the trot and give him a break of a couple minutes to catch his breath. Then do it again. Same principal - buck = smack, smooth canter = have a rest for a bit. Do this until he does at least 10 trot - canter transitions without even hinting at a buck!
The next time you get on him, same thing. Some horses I have had have worked it out pretty quickly and quit that type of behaviour after the first ride or two. Others take a few rides. The very first ride that you ask for a walk-trot-canter and he does all three both directions without bucking - tell him he is a good boy, jump off and put him away.
Stay safe and I hope you can sort him out!
All horses deserve, at least once in their lives, to be loved by a little girl.