Did the Democrats have a plan for the large population they intended to seize when securing modern America's southwest? The main point for opposing the start of the Mexican American War by Whigs was the "Spot Resolution", but I haven't found any arguments by nativists advocating against the war because of the large influx of Roman Catholic Hispanics. Was this a concern for the Polk administration?
The area that became the Mexican Cession after the war did not contain a large population (relative to its size). Part of the initial troubles over Texas between American settlers and the Mexican government (a significant precursor to the Mexican War) were due to the fact that the American population quickly outstripped the local Tejanos population.
In fact, one of the main arguments against annexing the entirety of Mexico, not just parts, was due to the perceived problems with suddenly incorporating a large new population into the fold (there are of course religious, cultural, and racial connotations to these ideas, but that's not the subject at hand).
As for the people who were in California/New Mexico, they were largely subjected to the forces of Manifest Destiny and American exploitation of the new lands such as securing Pacific ports and later, of course, gold excavation. One of the entities set up was the California Land Claims Commission (or something similar to that) which was meant to settle pre-existing land claims between Mexicans and newly arrived Americans, since land titles often contradicted one another.
A quick word on Catholicism, since you brought it up. Nativists did criticize the Polk administration when it allowed Catholic priests into the Army, claiming that Democrats were supporting the creation of a worldwide Catholic state. On the other hand, Catholicism could be used as a justification for taking all of Mexico. One Philadelphia newspaper derided the Catholic faith and so weak and inferior to American Protestantism that mass conversion would occur given the chance. They stated: "on general principles, Americans are bound to the diffusion of our own system of government over every enslaved people...In poor, degraded Mexico, even the first elements of civilization are wanting--and these are, respect for property, and the right to worship God according to their own conscience. To extend our free system into such a country would be the brightest stretch of mercy."
Examples from Amy Greenberg's Manifest Manhood, 99.