I've never been on the domestic side. But...(since no one has given you any advice)
(1) There are no lounges that I'm aware of on that side.
(2) The "good" news is that the line (queue, for the linearly-impaired) for immigration at 23:55 could be quite long, and so you won't feel bad that you are not wasting $ at a hotel (not even your strong CAD$)
(3) Your flight isn't the first of the day. I'd imagine by 2:30 the counter will be open, if it isn't a 24 hour counter.
(4) The Airport Hilton, Holiday Inn, or Sheraton doesn't exist, so you'd only have about a hour or two, MAX, to sleep, if you went to a hotel.
(5) The other "good" news is that LIM is no longer a nasty pit of an airport. Maybe a newspaper, a book, some coffee, or Inca Cola, will pass your relatively short connection time. I bet by the time you get your bags, go to LAN domestic check-in, and get through security, you'll only have 3 hours before boarding.