Having 1 ticket for MIA-LIM-CUZ only makes sense if you are planning to connect in LIM (a few hours) rather than stopover (more than 24 hours). Most flights LIM-CUZ are in the early morning hours, though cancellations due to weather from CUZ are higher than normal elsewhere. Most of the flights MIA-LIM are overnight flights so a stopover MIA-LIM stopover CUZ followed by CUZ-LIM connection MIA is very doable with 2 separate tickets.
All LAN intra-Peru flights are economy only, another reason to buy 2 tickets. Take LAN biz class MIA-LIM (767 lie flat seats) rather than AA's 757.