I've had the grand joy of taking that flight twice (EZE/LIM/SFO).
Lounge opens at 0230. Arriving at 0200 is plenty of time. The Y check-in has not appeared bad, although I've only been in the J line, and checked in w/in 5 minutes of airport arrival. Migraciones y security have been extremely rapid. Unless something unusual has occurred, your trip to EZE will be 30 minutes.
I generally have a bite, perhaps take a nap, finish cleaning up my place, shower, and head to EZE.
If I was a hotel-staying person, I'd probably keep the room for a pre-departure shower and decent bathroom facilities, and a secure place for all my belongings.