Is this recent behaviour or has he always been this way? How long have you had him and what's his history? What's your insurance situation, can you afford to throw some money at proper diagnostics?
I know you say you've had everything checked but the broncing, grumpiness and unpredictability screams pain to me (or perhaps, memory/fear of previous pain) and in my experience with horses this can be quite hard to get to the bottom of, and require some quite extensive vet investigations to get to the bottom of if the more obvious causes have been ruled out by the usual physio/dentist/saddler/lameness work ups. Hind gut ulcers for instance are quite common and only tend to be diagnosable with a full scope (not done on a 'normal' vet check), but you can always try the treatment first (gastroguard, ad lib forage etc) and if the symptoms improve that gives you a good hint. Likewise, kissing spines or some more subtle kinds of lameness e.g. hock arthritis may not show up on a basic vet check, you may need x-rays or nerve blocks, but if you do a 2 week Bute trial and he's better then you know there's something somewhere hurting him.
In the meantime I definitely don't really think you should simply persist in putting your DD up on what you describe as a 'demon pony' and hoping for the best, sounds well beyond the normal trials and tribulations of naughty ponies, surely if she keeps being thrown at best she'll lose her nerve and stop enjoying it and at worst there'll be a serious accident. I wouldn't just 'get rid' though, I'm not anti-selling on of child's ponies in the right circumstances, but what responsible home would want him as he is now? If you sell or even give away, you 100% will be risking someone else's child getting hurt or maybe a dodgy dealer buying him, buting/sedating up and selling on as suitable for a novice child esp if he's pretty which may lead to a spiral of being passed around from worse to worse home, and that's just not responsible or fair on the pony... If it was me I'd stop my DD riding him as of now, try the Bute/gastroguard trial (or perhaps an extended holiday and dose of Dr grass - not easy at this time of year though) and see if he's better on the ground and lunged first, then depending on results do more investigations to my insurance limit or perhaps consider reschooling and lessons with an experienced and sympathetic instructor. If no results and the behaviour doesn't improve consider retiring as a field ornament or even PTS. If that means your DD is without a pony of her own for a while that's unfortunate but I am sure you can supplement with lessons at a riding school or perhaps a share, and if she loves him as you say she'll understand the need to see if he can be made better before just dumping and moving on?