We are in the same situation, a late arrival and early departure the next morning. After checking we will likely stay at Hotel Costa del Sol Wyndham Lima Aeropuerto which is directly across the street. There is a 2nd story walkway or you can cross the street. When I spoke with the hotel if you have an international connecting flight, as we are going on to Santiago, Chile, you do not pay the $30 plus immigration fee to come back to the airport. I would certainly confirm that again. Also there is a new hotel complex being built with a 5 star Wyndham in the airport, but that is scheduled to open early 2025. Couldn’t get any details or specifics but sure would be more convenient to fly direct, but we couldn’t avoid that one stopover.