Exactly: EC261 only gives you a recourse against the operating airline (regardless of both airline code and purchasing process) if you have a valid ticket. BA claimed that you did not have a valid ticket and AM confirmed this, which seems to me to suggest that the EC261 avenue will be a waste of your time as you would - by your own admission - be suing the wrong airline.

My guess is that the cock up was strictly with AM because they didn't issue the proper e-tix so in my view, the issue is likely incompetence on AM side. In my view, complaint will be more effective than suing in any case unless you have too much time on your hands and do not care about stress. I would focus on the incompetence of not properly issuing the ticket and the unacceptable attitude of the agent you spoke to.