I wonder if the instructor knew that Simon might rear. If she did, then she should not have allowed you to lead him in. YOu are much too much of a beginner to be expected to handle a horse that has that strong of a reaction to being pulled on, by either lead or halter.
I do NOT think this is an issue of either this teeth or his bit. If it were, you would have had problems during the entirety of the lesson, which you said went very well. No, this is an issue of that school horse, like many such, knowing that once he enters the arena, he must work. He is reticent about going through. If it were just an issue of him not liking to go through narrow places, this would happen going through all narrow places.
My guess is that he REALLY does not like his job, and is protesting, mildly , at the gate. Then, when he is pulled to go through it sets up a dynamic of him wanting to resist all the harder It is possible that he has some issue with being hard tie (i.e. feeling hard resistance can trigger a feeling of panic, and the hrose pulls back , then rears up in automatic flight attempt. ) One would have to know more about his general behavior and history.
Anyway, back to his life as a school horse. Since you say he does well once in the arena, I think that he has not 100% soured to being a school horse. It's not an easy life for a horse, and I would not approach him necessarily with the same approach to discipline I might my own horse. He is going to be handled by a lot of beginners, so asking them to do any kind of complex moves it not going to work, at first.
I would suggest using treats as a reward. He comes to the gate, he gets a treat. You ask him to walk through the gate, when he comes forward, a treat. When he is in, a treat. Eventually, he will know that a treat comes from getting in, and you will only need to reward him once he is totally inside the arena.
Some will say this is trainning in bad habits. The bad habit comes only if he is rewarded with a treat when he 'mugs' you ( he snuffles hard into your pocket or hand or pushes his head into you demanding a treat). This cannot be rewarded. YOU must give the treat to him , never when he demands it by 'pushing' on you. Once he learns the treat only comes when he is inside the arena, he will walk in with relish!
As to putting pressure on a horse that is trying to back away. . . . yes, this can work,
but, you make it so that the pressure stays the same. If the horse backs away, you may need to back with him some, but you keep up a steady resistance, neither less nor more than before. IT's tricky. And when he pauses in his back up, you ease up on the pressure. YOu do NOT take up the slack he creates by him quitting to back away from you. You reward his choice to NOT back up, then you return with gently pressure to ask him forward. But if, as you experienced, the horse goes quickly into such a panic at the pull that it rears, then you probably need the help of a better trainer.
If that doesn't work, you use what @Foxhunter
described to encourage him to go forward, but, WHILE you have a gentle forward pull on the line, you use a long whip to encourage him to move his feet and when he does, if he moves forward, he creates his own slack, if he moves backward, you maintain the same equal pull until he changes his direction and follows that feel on the line, even the tiniest amount.