Just an update.... I was not able to interline my luggage between CUZ-LIM flight on LAN and LIM-ATL flight on Delta.

Instead, upon leaving domestic baggage claim in LIM, there's an area for luggage store. The service is relatively inexpensive and we were able to leave our bags there while exploring Lima during our 12+ hour layover. (Had a great time, by the way.) The luggage store area offer 2 options for bags: locker and open area (within the baggage storage office, of course) storage. You can put as much stuff as it fits into the locker for a single price, which worked great for storing 2 carry-on luggage + some misc stuff we didn't need for the day. In the open area, they charge by the piece.

I forgot how much they charge per hour for each of the 2 options...but I paid a total of US$25 for somewhere 13 or 14 hours of storage (1 locker + 1 luggage stored in the open area). They accept USD in cash, which was very convenient.