Is it possible to not check luggage? May avoid a headache for you. However, I'd say there is a reasonable chance there is an interline agreement; I would try to find out ahead of time...that way if the agent you get on check in isn't aware, you might have better luck convincing them if you are sure. If not, as recommended previously, consider collecting your bag in LAX and re-checking it onto LAN.
If none of that works,
this website, and
this website seem to indicate that passengers "in transit" are not assessed the fee. This is consistent with airport departure taxes and other fees I've experienced across Latin America in similar circumstances.