Well, I assume your flight is after January because LIM is closed nights until 7 am.

No, you cannot check luggage through, you'll need to collect it at international and shift over to domestic even if your had one tix... which helps with timing... you have immigration, waiting for your bag, customs, bag drop at domestic (they close 1/2 before flight), security, gate wait or lounge, but all pretty quick normally... so if the arrival is on time, you have time. If some problem, note that for say for Feb 8 there are 17 flights that day with 8:20, 8:35, 8:45, 9:05 etc. departures...

Of course, one FT trick is to go to the extreme right line at immigration, there are more agents looking after fewer lines, so much faster.