I doubt that BA would do this, but it's probably a reeasonable question. There are some stories of Ryanair banning passengers who use blacklisting, here is one such tale from Reddit but there are other examples out there. To be fair, this is customer contact staff in Ryanair trying to prevent people from using Chargeback or Section 75, what actually happens in real life, when Ryanair (and BA) run their reservations purely via computers which can be asked to provide and correct any personal data, is very much another matter. I doubt the CAA will be impressed, for example. In a somewhat different space, Nationwide decided to discontinue the bank accounts of some people who took advantage of a legal settlement relating to the way interest rate charges were calculated (similar to the one ongoing on BA for the data breach). The FOS ombudsman made it clear that their office was unimpressed by this outcome and later on a senior manager in Nationwide said it was the wrong thing to do.

In the case of Amex Chargebacks, I happen to know that several hundred thousand chargebacks have been entered against BA, and since they are usually processed by two separate locked down teams in India, Amex have received virtually no communication back from BA when chargebacks have been launched, so Amex has just paid them. I suspect this is a mess that will need to be sorted out at some point but I can't see the passenger being disadvantaged, BA needs Amex on its side more than ever at the moment.

Airlines would be well advised not to blacklist Section 75 claimants due to wider consumer protection on discriminatory trading.

Finally, what very little I know about when BA does ban passengers, it's to do with bad behaviour against staff. The bar is set quite high, in that just telling a BA staff member to get stuffed will probably be ignored, and even if you do a low level assault on a BA staff member - thankfully a very rare event - you would be looking at say a 2 year ban. A well known South London model with significant anger management issues (and previous form to boot) only got 2 years. So in that context I can't see BA getting too worked up over a few hundred pounds of chargeback, particularly if there was some basis to the claim.