What culture do you want the Caribbean halfway through the game to be then. Or the white population of Spanish America as a whole, who was discriminated against in favor of those born in Spain (note: I am aware this is a simplification of how the Spanish Empire worked. If you have a better idea on how to implement it I'm all ears.)
Well, New World cultures should be formable out of migrant European colonist pops and surviving Native pops to form new identities. To answer your question, pre-established historical New World cultures should form out of diverting and mixing European, Native and African pops in the Americas.
As an example, an in-game Spanish Cuba around 1650 should have Cuban culture pops who formed out of locations were Spanish pops (Castilian, Andalusian, Catalan, etc.) and native Taíno pops interacted. This Cuban culture would represent Cubans of Spanish descend (both "white" and mestizo) and have especial Estate mechanics including mestizo and criollo Estates, especially considering that you could legally "become white" in colonial Spanish America if you were a wealthy mestizo.
There should also be an Afro-Cuban culture (formed from African pops in locations in Cuba) as a culture and some Taíno pops still surviving in some areas were the control of the State is not as present.
This would also then apply to other regions in Spanish America.
Finally, Spanish American colonial governments should have a 90% chance of their rulers (viceroys and governors) being of an European Spanish Culture (Castilian, Galician, Andalusian, etc.), to further create tension between a, for instance, Castilian governor ruling over a culturaly Cuban and Afro-Cuban colonial subject.
In the case of the Americas throughout the scope of the game, we really ought to be seeing a lot of culture divergence and formations and nothing like the "unify culture" mechanic that this post is all about. The Americas must have a unique culture system, based on who colonizes which region, imho.