Ouch!
ba.com indeed starts out making it simple and clear "your personal allowance" is there for you in MMB. (as per links above)
So that's that in many ways.

The webpage then contradicts itself with talk of longest sectors and looking across to JBA partner Iberia ... where the link proffered doesn't mention the allowance!!
Not really sure I'd investigate any further pre-flight in that scenario... if MMB says 3, its 3.

Throwing some logic at this (where angels fear to tread). Its a ba.com purchase, on IAG via JBA skies. Why wouldn't they tell you the right baggage allowance AND incentivise you to mix routings by having the extra allowance on IB?

That said, it may well not be straight forward getting the refund.