Welcome to FT and the BA forum.

So is it the case that the booking in fact has separate fares - ber-lcy as the first booking, then another booking from lhr-Mex? Or is that a single through fare?

i wonder if codeshare might not be your issue here, but more that several different tickets with different fare rules may be wrapped into a single pnr owned by the ta because otherwise there would be no logic in adding baggage for specific segments. My sense is that to add luggage in advance, you might need to contact the ota who should be able to help though may charge you for the privilege.

Having said that, I would also add that almost invariably, adding luggage to a basic fare will likely be more expensive than booking luggage inclusive fare in the first place do wonder if you might have been better off doing that to start with.