As others have said, Mexico had no bargaining power by the end of the Mexican-American war. The amount of land taken was dictated by American internal politics, not Mexican-American relations or the fighting strength of the Mexican Army. America taking more land just requires Slave-owning expansionists being more powerful in Congress. If the Yucatán is successfully annexed, it actually would imply pro-slavery and more pro-expansionist factions had more power, and thus would indicate more land being annexed, not less. I would expect that most timelines with an American Yucatán also have more of northern Mexico seized by America.
Domestic impacts
While annexing the Yucatán as a state seems unlikely, it becoming an American satellite state seems plausible. Or a territory. However, if by some happenstance it happens, it does shift American politics pretty significantly.
For one thing, the center of gravity for the US just shifted south quite a bit. The navy has to patrol the Gulf of Mexico more, and trade between New Orleans and the Yucatán is going to lead to growth between both regions. Very few white settlers would move to the region, but a fair amount of white slave owners would expand into the area. And they would bring tens of thousands of black slaves with them. The army would have to spend a significant amount its resources garrisoning the region. Protestant missionary settlements would also prop up. This means the army and navy would have to be larger than OTL during the time period, but primarily focused in one region.
America would probably pull out of the region due to the revolts, it’s difficulty to govern, it’s separation from the mainland, it’s largely Catholic-Spanish or native population, and it's lack of commercial value. If it doesn’t happen within a decade, it would likely happen after slavery was abolished, whenever that would be ATL. However, if somehow it wasn’t abandoned, I could see it replacing the Dominican Republic as the place where America tries to dump all of its’s former slaves once they are free. In that case, you would wind up with an African American-Mexican Spanish-Mayan-Catholic culture with some hints of White American Protestantism emerge by the 1900s, which would be unlike anything OTL.
New Mexico was almost admitted as a state immediately after the war. I imagine to counterbalance the slave state addition, it will become a free state at the same time. It’s borders would probably be from either the Rio Grande or the Pecos River all the way to California. Taking with it all of OTL Arizona and the most populated triangle of OTL Nevada. This state would actually initially be majority Spanish speaking and have a large native population, and they would be somewhat better able to resist assimilation here. They would probably wind up a dual Spanish-English speaking state by 1900.
Foreign Policy Impacts
Central America was at the time in a low intensity fight for influence by the US and UK. America annexing a territory in the region is going to ramp up the intensity of the conflict. Expect the British to reinforce, instead of pull out of, the Miskito Coast and Bay Islands. Expect the annexation or Central America to be a more popular view among fire eaters, and more Americans to launch Filibustering expeditions into the region like William Walker’s attempt to take over Nicaragua for the US OTL.
Though still unlikely, a war between the two powers for control of the region would be more likely ATL. Britain would feel threatened by the growing power of America and her growing imperialist interests, and unlike OTL would catch wind long before America had parity of military power. And So long as Britain retains the Miskito coast, they probably build a canal through Nicaragua.
Also, expect more Americans to be in favor of annexing British Colombia or other parts of Canada to ‘balance out’ the Mexican Cession. With greater interest in Central America and a larger navy, the Danish Virgin Islands would probably be purchased earlier, expanding America’s influence into the Caribbean sooner. If somehow fire eating Slavocrats maintain their power, they may feel emboldened to annex Cuba as well. That almost happened several times OTL between 1840-1860. But it would be highly unlikely in a situation in which northerners have an earlier and more powerful disdain of slavery and it’s growth.
Abolitionism
Which leads me to the biggest change. The abolitionist movement would grow sooner and with more fury ATL. The Democratic Party just preformed a brazen and undeniable act solely to expand slavery and the interests of slave owners, over the interests of the country. The Yucatán offers little to no economic value for most of the US. It’s a large, swelteringly hot, malaria ridden jungle filled with hostile natives and Hispanic Catholics. It’s only value to the US is more pro-slavery congressmen. It would be incredibly expensive to maintain, and money that could be spent developing the frontier would have to be diverted to just maintaining this white elephant of a territory. The annexation of the Yucatán, especially if it’s combined with the annexation of more of Mexico, would radicalize a lot of northerners against the ‘peculiar institution’. And such a failure against the Slavocracy would kill the Whig party earlier and develop an alt-Republican party sooner.
Future Speculation
To get more into wild speculation, there are many ways this could go. Basically none of them are good. OTL the Civil War was all but inevitable by 1848. Here that is even more true. Other than that, here are some guesses.
1. A more brazen land grab in Mexico would plunge the country into a crisis for some time. We could see the country break down in the 1840s over it. Though I think that’s very unlikely.
2. This is what I think is most likely. The Yucatán is too expensive to garrison, and as compromise with the growing abolitionist movement it is abandoned by the early 1850s. Probably in exchange for something more manageable like Kansas as a slave state. A civil war still begins between 1856-1864, and while the south might be slightly stronger it probably results in the same outcome. I don’t think thinly populated territories in northern Mexico would change much, if anything. Essentially, The Yucatán Debate is another version of Bleeding Kansas and becomes the topic of trivia questions, forgotten high-school lectures, and alternate history discussions.
Mexico would be in no shape to retake it, so it becomes an independent republic most similar to OTL’s Belize and Guatemala. The Yucatán Republic would become a Central American state influenced by the British, then the US, and would abolish slavery because of American pushing after the Civil war. The only remnants of being part of America are a constitution and government very similar to America’s, a small Protestant community, a small African-American community, and many people who live in the coastal cities speaking English as a first or (more commonly), second language.
3. It could also butterfly into an Anglo-American war in the 1850s. Fire-eaters would want to expand slavery into the rest of Central America by booting out the British and Northerners would be more in favor of annexing western Canada to balance out the Mexican gains. This was a view held by a few like Seward OTL even. The British would want to strangle the nascent American Empire that in its crib before it grows too powerful. This merits a post of its own, but in either case of side winning expect them to dominate the continents. If America wins, a civil war would probably still happen, just later, in the 1870s or 1880s. If she looses, she probably breaks up in the aftermath.
4. Alternatively, the Yucatán could be maintained, somehow, and compromise could stop any Anglo-American conflict from heating up too much. Through a combination of luck, gumption, and a lack of organized opposition, the Yucatan remains a slave state until an Alt-Civil War between that would break out between 1856 and 1864. The Yucatán would have negligible impact on the war, as the Union navy would blockade it fairly quickly. At most the war extends by a year because some Union troops get bogged down there. It's biggest impact on the war would be decades later. Confederate apologists would inevitably attempt to justify that the CSA wasn't racist because a single Hispanic regiment fought in Louisiana or something. In reconstruction, the colony-state would either be abandoned or turned into a refuge for free Blacks.
If maintained, the Yucatan would become the previously described amalgamation of the already existing cultures with African-Americans. Such a state would be the odd duck of the union, and certainly the poorest. This would probably result in a very grim version of Jim Crow. In which African-Americans are forced to move to the state by the State or Federal government, either in some twisted belief that it would benefit them, or just in the South's desire to get rid of them. With little natural resources and little job opportunities combined with constant immigration, the state would be in a very rough shape. In the later half of the 20th century it could pivot towards tourism like OTL, but that's an unstable and imperfect industry.
That was a fun exercise, though I probably wrote a bit too much. It could make for an interesting TL.