I can empathize with the OP and BA's regimentation.

There's nothing painless in being told "no we can't" and then have to argue, and wait, in order to get them to do what they are supposed to have done in the first place [as per the facts in the OP] (it's the operating carrier's responsibility to reaccommodate you).

Pretty frustrating and a waste of time. And in OP's experience [as per the facts in the OP], BA's incompetence and aversion to customer service made him/her miss the flight he/she should have been painlessly reaccommodated on. Without argument.