Hi eze. I'm not the expert but I found a LATAM webpage about connections. For intl-to-intl transfers it says:

• Passengers will NOT have to go through the immigration and entry process or pick up their baggage if the flight is within the same reservation.

I think the key thing to realize is that, if you want to avoid immigration in LIM, you will have no access to an airline counter that can issue a boarding pass, and no physical access to your bags. If you have no checked bags, and a boarding pass (on your phone?) for the outbound intl flight that you can show, then I presume you should be ok. I can't see it working otherwise. I don't know that every airline will issue a mobile boarding pass for international flights. Many will want to verify your passport in person. And I don't think you can count on bags being checked through across separate reservations.

Your life will be easier if you leave at least 90-120 minutes connecting time between flights and just accept immigrating into Peru and leaving again. Assuming you're eligible for a Peru visa. Or of course doing the booking in a single reservation.