I believe LAN does ship bikes from SA to Mexico, specially as their cargo division runs a regular service from Santiago to Mexico City, however Im not sure if shipping with them is as easy as doing it with Girag.
Here are the details for MasAir, mexican company that also runs cargo planes and (i believe) are agents for LANcargo in Mexico
Teléfono: + 52 55 5786 9555
Fax: +52 55 5786 9543
Contacto: Mayra Sánchez
I havent looked that far into that because Im really trying to make it back on my bike, and ride the Sthlratte from Colombia to Jamaica, Cuba and land in Mexico (Isla Mujeres)