Agree with others, the more work you can do to get them both in balance the better, I have experience with ponies of this type and their confirmation does work against them as they aren't built for riding at all never mind jumping, you want them light in front and sitting/collecting behind to be able to properly use a half-halt whereas their inclination is to pull on a downwards incline leaning on their shoulders (and the riders' hands!). Also, the rushing (although easily taken as excitement/excess of energy) always used to be 100 times worse with mine when stressed and/or tired (and if not particularly fit they do tire more easily than you'd think, I think with their thick gullets they can also forget to breathe properly), my little cob mare although a surprisingly good jumper for her type would curl herself up into a little ball of anxiety and tension and become a horrible ride into a fence as a result.
Why can she only have lessons in a small arena, I think it would be really helpful for an instructor to see what's happening in a course and coach her through it a bit, can you look at taking them off site if current yard only does limited lessons, does she do PC? Ideally with her jumping you probably want to look at pre-empting the tanking, so don't just set up a course, send them off and expect different results, build up more gradually and use lots of placing poles and grids to sort out the striding and fix the rhythm a bit for her. With ponies that get overly anxious or excited jumping it can be helpful to have fences up in the school and occasionally pop one in a small/low key way as part of schooling, then go back to flatwork so they don't equate jumping = tear-off every time (and accustoms the rider to the odd fence not being a big deal either). I wouldn't stick only to single fences but perhaps build up more gradually to courses, so do short courses of maybe 4 fences - it's easy to fall into the trap (esp if you're limited on facilities/space/time to set up) to only ever jump 1-2 fences at a time when schooling then suddenly leap up to 10-12 in a row when competing or at a lesson, but that must be confusing for the pony and like I say I think fitness (rider and pony!) is an often neglected issue, if they rarely even canter for more than a lap of the school at a time it's a big jump (pardon the pun) to then have the balance and puff to go for 2 mins solidly with fences in between (more if there's then a jump-off or if the teacher says 'go again!')
Schooling wise it's probably the same old thing you always need to do 
, although helpful to have a diagnosis/tailored exercises from an instructor, lots of transitions focusing on teaching him to respond to the seat and not just the hand, and encourage him to try and 'sit' through the transition not just collapse onto the forehand. Lots of school shapes focussing on rhythm, balance and straightness/bend, basic lateral work, raised poles and cavaletti are great for getting some more push from behind. Lunge work and without stirrups would be good for your DD and her position/seat.
There are some good suggestions on this thread which asked a similar question a few weeks ago albeit a different type of pony:
https://www.mumsnet.com/talk/the_tack_room/4837690-bit-advice-pleasewhat-to-try?reply=127525698