Lima and much of coastal Peru is said to have this nasty sea fog and/or permanent overcast all winter (9 months) if the Humboldt Current is particularly cold.

That said, I would think the highlands first would make more sense if you have to absolutely make your flight home. LAN has LIM-CUZ flight starting at 0500 so you're no wasting too much time at LIM (immigration/custom lines are said to be bad in the early morning when the N. American flights arrive anyway). That way, you don't have to spend that much time at LIM on your way home.

That'd put you into CUZ before 0630 and unless you're coming from high altitude, I'd go to Ollantaytambo or Aguas Calientes that day (much lower in elevation than CUZ which is 4-5k "higher" than an aircraft cabin is pressurised to). You can get a car that will take you to see the sights along the way.