The one time I had similar EZE transit, I was flying UA to LA - I went to the AA lounge (there's no LA rep there) and the AA agent called the LA desk, and they checked me in, and told me to get my BPs at the gate prior to boarding. (There were no attended LA gates when I first arrived). This was a few years' back so I don't know if it is any more advanced or not, but I'd head to the AA lounge first.

As an aside, make sure you are armed with your actual ticket numbers issued (and the latest updated ones if there have been any reissues since the initially-purchased ticket), especially if LA was NOT the issuing carrier.