Could be a variety of things, one quick check is split them up, does the same thing happen with same pony different rider and different pony same rider? It's quite possible even if she's very confident that your DD is tensing up when jumping in competition and that's sending anxiety signals to a sensitive pony. Plus of course it's a vicious circle, if DD has prior history with the pony playing up in that particular circumstance then she'll naturally be half expecting it again and that will make her ride differently, if there's something making pony anxious about the environment or the atmosphere she'll be even more tense and will feel different, more backward before she's even started which will fuel nerves etc. Good news is if it is just that it's very overcomeable, just needs more time and lots of very easy confidence building outings, I'd do low-key unaffiliated clear round type show-jumping with very, very small fences and no fillers, potentially even just take her in the ring, trot or canter round in a nice forward rhythm then pat her and take her out without even asking for a jump. Or do very very small courses so she can just walk over them. Only once they're both fully confident with that step the height or difficulty up. You can continue working on harder things at home or in lessons but for competition keep expectations as low as humanly possible.
To sound a slightly more ominous note however - before you do any of that I would give her a full physical MOT and check over, saddle, back, teeth, physio, vet, feet - I've dealt with so many kids ponies over the years and what you describe of the pony being fine at home, fine on the flat, fine at polework clinics and PC rallies, but get them into the arena for competitive jumping and they just don't want to do it any more, sudden napping and dirty stops and so on in a previously very reliable pony. Sorry to say in 9 cases out of 10 it was a physical cause, not necessarily obvious lameness or unfixable terrible issues but something that was hurting them enough for them to say no to competitive jumping. Usually the pony was basically honest and a nice sort and was willing to try, particularly at home and on the flat, could often be bullied around a course by a stronger rider, but get them in an unfamiliar environment, away from other horses, different surface and new/scary looking fences, rider a little tense or uncertain and it was brakes on, no thank you, not today - if you think there's even a chance she's in discomfort it's definitely worth ruling that out before simply saying she needs stronger riding... And I have to say I'm always a tiny bit suspicious of any pony making it to 12 and still being very green, not to say there aren't genuine reasons but it makes me wonder whether the pony's been previously badly started and frightened itself/been left in a field as a consequence, and/or had a physical problem when in work that could now be recurring. Not saying this is definitely your pony, just to be on the alert...