I really don't think you should be spending £7K on a pony if you're not going to affiliate and go all the way, as previous poster said it's a lot to loose if he breaks.
There are lots of pony club ponies coming up for sale all the time - a really lovely one (who was on the area teams for sj, dressage and XC, and regularly compets at 1m ) has just gone on the market in our PC for £5K, he is a tall looking 14-14.1hh. I say tall in that he seems bigger than our 14.05h New Forest - who was cheap and yet is also on the team show jumping, regularly double clear as he's very careful, and never refuses these days.... are you sure it's not the rider rather than the pony?
We found ours started refusing big style after daughter lost her nerve about some scary flller or other. It took a LOT of coaching, and recuse remedy for her, along with calmer for him, as he used to fizz up and get a bit unmanagable and dive out at the scary fillers - horses are herd animals and need a leader, and a nervous leader transmits her fears to the horse. Could be a bit of an anticipation element - does she fall off when he refuses? - and he picks up on it thus DOES refuse. After another 6 months of competing where I'd dropped the pony calmer but still continued with the recue remedy for her, as well as telling her the pony had had his, she overcame her nerves (ridiculous she was nervous really as she'd NEVER fallen off him apart from when he slipped over once on a circle in the mud) and with some excellent tuition at Pony Club (and she really is a dedicated instructor, last time out took 4 teams the teams finished 1st,2nd,3rd & 4th in the team class, and completely stole the show in the individual results) we found the pony flew round everything. Now they're qualifying at 90cm without calmer or rescue remedy - and daughter is asked to go and ride naughty little ponies who refuse jumps.....she's firm and insistent and holds the line till they don't refuse, and then their younger rider gets on and flies round.
You say you feel novice - just wondering have you had an older/more experienced rider take the pony jumping, and have you had a different instructor (ideally a show jumping specialist - look for recommendations locally and be prepared to typically pay £40 or so for 45 mins ) to look at the pair and give them some lessons and improve their technique?
Also has the pony had the usual feet, back, teeth, saddle checked in case there any issues meaning jumping higher is uncomfortable?
Finally, not being funny but at 13 she's likely to be growing fast and you may need your 7K for a horse soon! (we do, think this will have to be daughter's last year on ours and she's 13 1/2)