If you are OW Sapphire or Emerald (AA Platinum or Executive Platinum), you should be able to access the lounge in LIM unless the airport is split across terminals and there's no lounge in your departure terminal. You can visit http://www.oneworld.com for information and locations of specific lounges throughout the system. All the rules about access are there too.

If you have no status with AA or LAN however, I don't think you'll be allowed to enter the lounge since your onward flight is in Coach. An exception might be made if the LIM-LPB flight is one-class and you arrived in Business; you can ask when you check in at LAX.

In either case above, I believe you'll get lounge access at LAX in the shared OW lounge at TBIT.