A couple of things here.
Basic Economy for elites works different on AA and BA.
With AA basic economy, elites on basic economy can not select a seat in advance, but do keep the free bag benefit.
With BA basic economy, elites keep the privilege of selecting a seat BUT have to pay for bags.
Since the outbound was on AA flight, it was expected that AA did not charge you for the bag.
On the return, I assume that what the BA agent saw is the BA policy, and even as elite, you have to pay for a bag with BA on basic economy.
I understand the discussion of what the ticket says etc, but I have always thought that when it comes to baggage allowance, the operating carrier rules are the ones that prevail.