Unfortunately, I don't think your situation will allow you to pursue a negligence lawsuit in a cost-effective way. I would suggest you obtain whatever orthodontic care you think you need and get a second or even a third opinion if need be in order to satisfy yourself that you have done the best you can with your orthodontic needs. This is just my opinion, of course, and you should contact other attorneys for their opinions on this before you make a final decision about whether to proceed with litigation or not. good luck to you and your family.
In order to prove any case of medical negligence the law generally requires that you have another physician in the same specialty as the one you are suing testify that malpractice occurred and caused you damages. Before attempting the legal route, I would bring the progressive digital photos to another orthodontist, who uses the incognito braces and ask for his second opinion. If he says that you needed that bracket, or something else, then the damages are likely limited to the approximate 1/2 year lost, which would likely not be significant to justify bringing a very expensive medical malpractice case. If he agrees with your current dentist, you can get another opinion or accept this opinion. If you accept it, I would continue to document the progress, or lack thereof, with photos. Make sure that the second opinion orthodontist does not know your current one so that you can get a truly objective opinion. This might require getting such an opinion in another city or state. Good luck and I hope it all works out for you.
It would be pretty hard. Medicine and dentistry & related fields are not an exact science. It is doubtful that your orthodontist made any guarantees about the outcome & success of your braces. I would go to another orthodontist & get an opinion about what your options are.