Not sure that it will get to a law suit. If you can show the backup paperwork reflecting that J was advertised and sold on the B6 segment, it should be a relatively easy chargeback (if the threat of the chargeback does not result in a quick refund). Outfits which have high chargeback rates tend to do a lot to avoid even more chargebacks and these guys likely have a bad rate.

I would just be careful that the new ticket is not markedly more than you are willing to pay before you attempt all of this.

However, before getting to that point, it is worth a shot at doing a three-way call with the TA and B6. Neither are going to be particularly happy about this, so just be polite & firm. Your role is not to argue about anything, but to simply keep asking what needs to be done to assure that you are seated in J (MINT) on the B6 segment.

Do this when you have plenty of time on your hands as well as a speaker with a mute button and something else to do while you sit on endless holds.