Looking at individual bookings using MMB sometimes doesn't give the correct answer. I think that this may be because it picks up "2PC" from the ticket, but doesn't correctly pick up the additional weight allowance. It's possible that these extra allowances (both the extra bag in economy and the extra weight per bag) are non-ticketed policy allowances. We get questions about this quite a lot.

However, check-in agents ought to know this. I very much doubt that any check-in agent in London would have a difficulty with what allowance you ought to have, but one of the issues with outstations is that local handling agents' staff may not always be as up to speed with the detail as they ought to be.

In your position, my first priority would be to make sure that the bags travel, even if the ultimate outcome is that I have to pay. I would then be chasing BA for a refund as soon as travel was over.