The ticketed allowance for baggage should be applicable for all flights on that PNR, no matter what class of travel.
Extra due to status with the airline you are flying can be different. More so if flying on an airline where you do not hold status.
BA & AA have a close joint venture. so status benefits tend to the same.
Many USA airline staff seem unfamiliar with tickets that do have baggage included. More used to charging $$ for checked in bags.