Generally the answer is no, LAN rarely lets anybody check in before T+4. MIA might if they're even there (I've had to connect from DL and they didn't show up until about 20 minutes after check in should have opened) and I've had EZE let me on a couple of times or offer to send me though SCL if there was an open seat in J. The EZE trips were before the new lounge and I had a little bit of trouble checking in at T+4:30 a few months back so it's hit and miss. The given reason that they don't want people checking in early is losing baggage. There might be an issue at MIA since LATAM has been mixing the agents between the LAN and TAM counters so who knows what will happen, I've had the nice TAM lady having to ask what to do about the lounge and baggage allowance when a Comodoro is flying in PB down to BOG.
LIM is not the best airport for connecting because of the ground staff there being a royal PITA with both real and imagined rules. You'll be connecting during the late night/early morning bank of flights from all of the other stations so it can be kind of a zoo, any serious delay on your inbound can leave you with a 12+ hour layover if you're heading to the USA ( or being stuck on an Airbus 320 if they route you through UIO or BOG to MIA). The SUMAQ lounge isn't all that great but it's there.
You might be able to get checked in if the LAN people are there at MIA since you're a Black, and I've seen the Ladies in Red a couple of times in LIM getting Blacks through the system.
Of course with LAN YMMV!