Canter transitions are hard especially on a riding school horse so you have my sympathy. Riding school horses are often used to being rushed into canter in a pretty rough and ready fashion, different riders will ask slightly differently in terms of leg positioning and with novices their leg aids are usually not particularly precise, the difference between a squeeze to say trot faster and a canter aid (and a 'whoopsie I got unbalanced and my leg swung backwards, I wasn't asking for anything at all!') isn't very clear, leading to confusion for the horse about whether or not you actually want canter, and of course being the saints they are, the default if confused is usually slow down/stop rather than spring off into a lovely forward canter which will unnerve people if unexpected! Plus it's not uncommon for the rider to get unbalanced during the transition itself and pull on the reins or bounce on their back which is not particularly nice. So I expect the reluctance to go and hollowing is probably related to all that, but it's not super comfortable to ride for sure.
What can help, practice, a lot of it, I know that's trite/annoying to say but it will just come with time and building your strength and balance, anything you can do to develop your lower leg stability and depth of seat (which can be done in the walk and trot, you don't need to repeat lots of canter transitions although this helps too), so no stirrups work, lunge lessons are great (can you sometimes use your private lessons for a lunge session?), or do you have someone who does mechanical horse or horse simulator lessons nearby as these are brilliant for position work?
On this specific horse I'd think about what mechanically is going on for him to try and achieve a smoother transition - do you ever see others ride him? Observe the canter transition carefully if so. You want him to push off from behind (his inside hind specifically to strike off into true canter) and lift his front end for a nice uphill forward gait. I suspect like most riding school horses he probably lacks impulsion and doesn't tend to work correctly over his back. So although yes bend and suppleness is important and you don't want hollow giraffe, pulling his head down or from side to side to get 'outline' or bend probably won't help, he will be more likely to tip onto the forehand or go behind the vertical leading to an unbalanced or rushed transition. What you want is suppleness over the back and through his ribs and core rather than just head down. You can experiment with exercises like asking for a controlled step or two of leg yield from your inside leg then ask for canter, do it from the 3/4 line towards the fence or on a circle but aim for literally a step or two only, as he lifts his inside hind up is when you want to cue the canter, takes some thinking, timing and coordination from the rider but can help the horse get the idea of stepping upwards into the canter. Or try asking from your light seat, again may help him feel confident to lift his back through the transition.
Generally I'd say the single biggest problem is rushing the transition, I know I do it myself particularly if it's a situation like a dressage test where you really want the transition to happen at a presice point, you start with what you think is a nice quiet but correct aid and from a good position, the canter doesn't immediately happen so you start tipping or flapping your hand or booting, horse rushes off in a flat fast trot unbalancing everything further and it's all a vicious circle - try and pay attention to the quality of the trot, plan your transition, get yourself well balanced, sit deep, be confident you are asking correctly and just give it a beat to come, if you don't get the canter first time you can back up with a flick of your stick or simply come back to the nice balanced trot and ask again - getting flustered is the enemy of ever getting it correct!