This is the unfortunate reality of running a business in a highly-regulated space affecting multiple regulators, e.g. US and UK.
For the same reasons that people often come to FT wondering why a hotel did not siply apologize and comp a night at the hotel for some minor injury, hotels would say that it is better to sustain the occasional loss than to admit fault and risk the one crazy person who sues and obtains a lot more. Thus, everybody gets the same treatment.
The people handling this issue at BA are low-level regulatory paralegal types. They manage risk according to standards they are provided. They are neither heartless nor uncaring. They simply have instructions.
While it is true that OP may now file various complaints and legal actions and that those will likely cost BA more than resolving this with OP directly, BA looks at the transport of millions of passengers a year and simply builds risk into its model. OP may be persistent and act. But, for every OP, there are however many more or who do nothing even if they first complain. Over time, the law of averages cuts heavily in BA's favor.
For example, there was an academic study several years ago which suggested that less than 2% of likely valid EC 261/2004 compensation and refund requests were ever even made in the first place. That number drops off as rounds of denial continue.
I agree that at this point, OP should cease discussion with BA and make a decision. Either drop it and move on or file complaints on both sides of the Pond and pursue an MCOL complaint for the various compensations and refunds due.