The “no checked bag” in MMB is definitely a(nother) website glitch.
As for the correct checked baggage allowance, since AA is the first marketing carrier, for this itinerary AA determines the baggage allowance, although it can apply whatever allowance it wishes (eg the BA allowance if it differed). Both carriers have a standard 2 bag allowance for WTP/PE, and 3 bags for OWE.
The query item would be the weight per piece. BA allowance for Gold/OWE is 32kg per bag in any cabin (at least for your baggage allowance on that flight, not sure if it applies to additional pieces). AA allowance in PE is 23kg (32kg in J/F). Not sure how BA check in staff would handle that one.
What does the e-ticket say about baggage allowance? That will take precedence over anything in MMB.