So we've arrived back in the States now, and I have a reliable internet connection so I'll report what happened. At GRU, AR wouldn't let us check in without paying the reciprocity fee, however, they offered us a partial refund to go check and see if another airline had a direct flight. We checked, and concluded that the reciprocity fee would be cheaper than switching airlines. We payed the reciprocity fee, and boarded the flight. Once we got through immigration at AEP, we had already missed the 16:00 bus, which by the way, took us forever to get tickets for. (We went to one kiosk who gave us boarding pass-like ticket cards, and we were told to go to another kiosk to get the actual ticket.) We finally got on the 17:00 bus, which makes one stop in the city, getting us to EZE at 18:15. After checking in at the seriously understaffed checkin counters at EZE, we proceeded to go through security and immigration, to finally board our flight at 19:05. The agents at the gate were not too happy about us delaying the departure of the flight by a couple minutes.

All in all, it was a hectic experience. We had to pay the reciprocity fee at GRU, miss the bus, and almost miss our connecting flight to get to our destination. Fortunately, our bags showed up on the carousel at LIM, and we made it safely there. Lesson learned, fly TAM/LAN next time directly to LIM.