One thing the acquisition of more land does is to reduce the immediate political tensions in the US. When California has the gold rush and is admitted into the Union, there is a much different Compromise of 1850. Cuba might be admitted as slave state so the South doesn't get some other things. There may be no fugitive slave law. Slavery is probably prohibited outright in the Utah Territory. New Mexico Territory and any other territory in the additional Mexican lands may get a form of popular sovereignty where slavery might be instituted.
There is likely no Kansas-Nebraska Act. With slaveholders seeing Mexico as possible expansion, they are likely not going to risk overturning the Missouri Compromise. That means the Whigs likely stay together and the Republican Party does not form, or at least not yet.
If there is a Civil War, it likely gets delayed by several years. And the longer it gets delayed, the more advantage the North has over the South, and it becomes more and more apparent that it is a very strong advantage which is likely to make moderate Southerners less likely to support seccession.
Most of the new Mexican lands are not conducive to larg scale plantations. Even if they are admitted as slave states, those states are more likely to be similar to the pro-Union Border States where the bulk of the population is not sympathetic to any Confederacy. So let's say Kansas is admitted as a free state in 1860. Southerners might demand a new slave state to be added at the same time or very soon.
What are the candidates? Maybe a state of "Rio Grande" that combines Coahuila, Nuevo León, and Tamaulipas. It'll be similar to Texas, but unlike Texas won't have any native slaveholders. It'll be immigrants from the South, perhaps those who IOTL moved to Kansas-Nebraska. This creates the political crisis because there will be a large number of people in Rio Grande who don't want slavery. Thus we may see the rhetoric and compromises in the 1860-1864 period we saw in 1850-1860.
We know how that will fall out. The Whigs collapse and are replaced by free soil Republicans. The native Mexicans become a large component of this Republican Party, giving the party a solid basis in the Mexican territories and Rio Grande. This totally throws out the existing strategy of the Southerners. Mexico won't be the key to preserving the institution of slavery. Instead of reliable pro-slavery votes, the situation is at best much more complicated. At worst, future Mexican states might be anti-slavery.
The crisis reaches a head in 1864 or 1868 when the first Republican is elected President. The question now is without the circumstances of 1860, does the same domino chain hold? South Carolina will still want to secede, but unless additional states join in, we won't get a war. Instead, we have the same crisis in 1832. There is a very different balance of power now. The Transcontinental Railroad is being built. There is likely strong support for the Republican Party in (American) Mexico. The "North" has even more economic and industrial power than it did in 1860.
If there isn't an effectual President like Buchanan in office, how successful would be any attempt to seize Federal property? What if the incumbent President takes a very strong pro-Union stance by making sure commanders of Federal troops in the South are not Southern sympathizers, but strong union men? Or gives orders that soldiers are not too surrender any arms or arsenals without express permission by the War Department? A strong stance, something a President like Andrew Jackson or Zachary Taylor would have done, would send a chilling message to the South. It might enrage the fire eaters even more, but it will be the moderates who decide what to do.
The Civil War might be avoided entirely as moderates keep delaying a showdown and every year that passes establishes the economic dominance of the "North" and strengthens the anti-slavery forces. Slavery could last a lot longer, but is slowly overturned as some point paid compensation for abolition goes into effect.