Update: How to get the best rate with LAN...
After 10+ phone calls with LAN I finally resolved the mystery around their AirPass program.
OneWorld Visit South America Pass and LAN South America AirPass are two seperate programs which have nothing to do with one another.
In my case, 4 legs (Buenos Aires-Iguazu-Buenos Aires-Bariloche-Buenos Aires) were $624 with OneWorld and $757 with LAN AirPass (incl. taxes and commission).
How to Book OneWorld AirPass?
This is the tricky part. If you call LAN they will try and book for you their own AirPass, not the OneWorld one. You need to make it 100% clear to the sales rep that you want the OneWorld ticket and not the LAN Pass. If they say it's the same one, they either don't know what the OneWorld pass is or they're trying to sell you a more expensive ticket. The OneWorld ticket is V-Class whilst the LAN pass can be L-class or B-class.
Once you get to a sales-rep and make it clear you want a OneWorld card, they will take your OneWorld electronic ticket number to/from South America and get back to you within 24 hours with a quote.
The only case that I found the LAN pass to be cheaper is if you fly to South America with LAN (as opposed to with a OneWorld partner). I flew Iberia as it was $400 cheaper than flying with LAN.
I could not find any other information on the web about the difference between the two passes. If you find my post useful, please comment!