So, one thing that's not been mentioned is if the introduction of Mexico to the Union means a dramatic shifting of economic patterns further south to the point where you're ironically dramatically strengthening the Mexican heartland in a way that makes assimilation and long term stability harder to keep. Mainly because, in essence this just kickstarts the Porfiriato 30 years earlier. Another good example of comparison might be the Mineral Revolution in South Africa.
Lets assume the Mexican liberal collaboration holds out at first and Mexican conservatives and large landowners are co opted through Foreign Direct Investment that lines their pockets enough that they can cast their ballots with ruling American elites for the time being and make the transition into Argentina style big landowners who also start buying into industry with new industrial barons on the bloc joining the elite circle. Stability rules for the next decade and change, probably more.
Ok, what now?
As others have pointed out, if we have the addition of Mexico it means economic patterns swing south to the point where the Deep South has more industrial pockets than before (and is now linked to Mexico's labor force/markets/etc) who can now use their capital -alongside European investment to jumpstart the local economies- then Mexico as a region is going to SO MUCH wealthier to the point it will dramatically impact America at large. We can surmise that the basic extraction patterns will be established:
-the restarting of silver mining (now with modern techniques). This is especially important.
-the formation of vast new cotton estates in the Northwest- which may be owned by Southern slaveowners using slave labor draggled along- or not; considering local and immigrant labor will be available soon enough the instant rails are established with the heartland further south that's full of peasants eager for more wages
-in general, tons of newly opened mines (Pacific Mexico is full of minerals).
-cash crops; mostly in the Yucatan but also in the Gulf Coast. Off the top of my head, this includes coffee, henequen, chicle, and sugar.
-industrialization of the Valley of Mexico proper as it has the largest population and densely packed cities (and will soon become a major center for remittance flows)
-a boom in rail development in order to get extractive industries to their buyers up in California, the American South, and the Northeast
-port investments for similar reasons
-all the various secondary economic activities to support these major ones such as housing, luxury goods, clothing, and so forth.
In short, these are all high profit, and extensively labor intensive economic activities. There's going to be immense demand for getting the rural peasantry and any immigrants into these sectors (and therefore incentivizing them to learn Spanish and assimilate into Mexican culture), for hacienda owners to modernize themselves and diversify the economy (given the Porfiriato's experience, they did this quite well), and for extensive immigration to come south into Mexico to meet the demand. Immigration from either the Caribbean, Asia, or Europe. (I suspect mostly the latter two). Combined with the end of decades long warfare and the assumption of Mexican debt into the US's name, and this frees up so much extra capital for Mexicans themselves to invest in. In short, a dramatic economic boom thats going to feed into a massive population boom as wages increase, food production increases, the death rate decrease, more people move in, and the demographic transition kickstarts. It would not surprise me if Mexico reached its 1910 population by the late 1870s (basically from 7.5 million to 15 million, and quickly surpassing the Confederate population for comparison)
You might wonder, well what does this mean for slaveowners who would likely move their slaves west? It looks good at first, but it really isn't. Slavery is incredibly expensive both in maintaining people's food/clothing/medicine/etc, and buying (and now the price of slaves will skyrocket since there's so much demand for it out west but importation is negligible). To say nothing of the security costs and ensuring slaves don't escape south or north. Immigrants and peasants are going to be in the long run cheaper due to less maintenance costs (and probably start up costs in transferring all those people over). To say nothing of using non slave labor just looking better from a marketing standpoint.
I don't expect tensions over slavery to diminish, now there's an additional front of wage workers who will dramatically dislike competition from slaves alongside the various moral arguments. If anything it'll probably flare up as before and the American Civil War pops up. From there, it depends on how big Mexican nationalism is. My money is it'd come back in a big way, since Mexico proper would have been knitted together into a functioning unit in about a decade and a half of all these changes.
Come the war, you might get a 3 way civil war with the Confederacy in worse shape. Incredibly hard to say after that, given all the potential variations.