My first reaction reading your post was that there's something off about a 14 y/o Arabian who doesn't "want" to do anything more than a slow walk; the story doesn't add up. The barn where I ride has a therapy program, and with the exception of the older horses, all the therapy horses MUST be ridden by non-therapy riders and given a more challenging job or they get mean and cranky and otherwise unsuitable for therapy. It sounds like the seller's daughter had been riding him, and I doubt she would have been happy just walking if she's old enough to start barrel racing. Arabians are also stereotypically a 'hot' breed and I've never met one yet who was content to plod around at the walk. Did you have a pre-purchase exam where the vet verified the horse's age (might be older than stated) and body score (might be skinny/undernourished)?
I think the horse is doing his best to test your son. He found that simply not moving worked for a while, and then when he found that he HAD to get moving, he tried bolting. It sounds like he probably just needs a tune up from a good trainer- a trainer who will actually ride the horse first and find out how he responds to a bit, whip, etc.
I don't understand why the trainer you worked with insisted on having the horse bitted. Beginners aren't as aware of what they're doing with their hands and what is a proper amount of pressure, etc. so I firmly believe they should ride bitless until they can demonstrate that they know how to properly handle the reins. The seller said the horse went well in a bitless bridle- did s/he say if the horse had ever even used a bit? If the horse was always ridden bitless, or had been a very long time since being bitted, or the bit the trainer used was too harsh or ill-fitting, then that would explain the head tossing and bolting.
Do you have an enclosed area to ride (arena or round pen)? I'd stick to this area when your son is riding for now. If you know how to lunge with a rider, that can help build his confidence back up. Otherwise, I'd stick with him riding during lessons only until the horse is sorted out.
I'd also either send the horse to a trainer, or have one that can come to you a few times a week to work with the horse. Be sure to get references, ask questions and think about whether the answers really make sense or not. I don't think this horse needs any strong training aids or gadgets- just some miles under an experienced and sensitive rider.