Within the past year I have been in all three lounges. In my opinion LAN has the best lounge, although I will certainly give AA a great deal of credit for a vastly improved lounge with good food offerings.
Delta uses the Aerolineas lounge which in my humble opinion is really third rate. The food is nothing to brag about and when I was there (I believe twice) the parents didn't give a darn as to what their children might be up to. The remodeled airport is quite nice and I wouldn't be surprised if the same firm didn't do Mendoza and Montevideo.