Aeromexico has some excellent fares available in business class from New York to Buenos Aires, with prices from $1733 round trip when searched directly on the Aeromexico site. No Mas Coach is also reporting fares from the west coast starting at just over $2,000 to Buenos Aires or $1920 to Sao Paolo. Note that the lowest pricing requires an overnight connection in Mexico City — shorter connections are available for a couple hundred dollars more.
I could not find these sale prices through Google Flights or other US-based OTAs, though I did find $1733 round trip from New York to Buenos Aires in business class via OTAs at Momondo:
Availability on the outbound looks good on Aeromexico’s site, with prices for the outbound leg from $828 mostly on Tuesdays and Wednesdays:
The return trip to New York was more challenging in terms of availability, but I did find it bookable beginning mid-week about two weeks after departure from $904:
The flights between New York and Mexico City are on a 787 Dreamliner. One Mile at a Time has reviewed Aeromexico’s 787 business class product, and it looks pretty nice. Between Mexico City and Buenos Aires, Aeromexico flies a 777. While No Mas Coach reports that the 777 product was also comfortable, it looks like it has angle-flat seats.
Still, round trip business class to southern South America for $1733 is an excellent price. These flights should earn 125% mileage when credited to Alaska Airlines:
Based on a flight distance of 13,349 miles, this round trip should earn just under 16,700 miles before any elite bonuses.
The Aeromexico site did not have a calendar view flexible date tool, so finding this pricing may just require trial and error by playing with dates. The fare rules seem to indicate that a stopover is possible if you can figure out the Aeromexico multi-city tool.
As with all airfares sales, prices can change at any time. Best of luck finding dates and availability for your trip.