The Gringo tax is included in the ticket price. LAN is the only entity with the predatory pricing, which is why the local budget carriers seem so much cheaper to you, but may not to residents.

If you book on the Spanish version of the website and select Peru citizenship (and can produce a resident card), you will be offered many of the tickets you are seeing for $200-300 on the US site, for $99. You may have to be on an in-country computer to do this, I haven't tested it.

They do check ID.