A border further south is possible, especially one avoiding more-populated areas, but the OP said "the border states". The
population data for the border areas at the time shows about 300k in Sonora and Chihuahua combined (excluding the part that was annexed in 1848), 300k in the three eastern states. The Western states are marginal land, but most of what it is practical to farm is held with solid legal title. Not an attractive destination compared to the vast expanses of unfarmed land to the north, you don't get anglo majorities until much, much later than the civil war, perhaps never. Baja is easy though, as you said. I simply can't see any of the conquered territories being admitted as states, individually they would be two powerful in congress, united they would have too many presidential electors. And we have to remember, Americans at this time considered Mexicans both racially and culturally inferior.
Mind, the only way I think any American annexation of the "Rio Grande Republic" states would have to grant them a state status in the short run. They are too highly populous to prevent from becoming states, in any such case. And, while definitely Mexican, those three areas hold, I believe, the highest concentration of European-origin Mexicans (rather than Mestizos) outside of the Mexican metropole in the southern part of the country. And, to the US at this time, that would matter.
It'd be a comparison to Cuba - which, while having the highest concentration of European-origin Spanish speakers, was also far, far more populous than even an extended Mexican cession, and was certainly looked at as governable and possible for integration into the US. We're looking at 900k for Cuba in 1850, half again as much as any further Mexican cession, and it was certainly considered for integration as a (slave) state. Cuba is far more profitable, but the possibility of even more slave states than just a single one in Cuba would be very intriguing for the Southern States. Note that it was two southern Senators who proposed altering the Treaty of Guadalupe Hidalgo even further south - Davis proposed including the south bank of the Rio ground and an earlier Gadsden, while Houston proposed pushing it as far south as San Luis Potosi.
I'd still stand by opinion, with a "Rio Grande" State entering as compensation for splitting California rather than permitting it to enter as a single entity, but with Southern efforts to attempt to subvert the Rio Grande with both settlers, hustlers, and other politicians in order to secure their electoral and senatorial votes for themselves. You would likely see "Rio Grande" statehood be delayed in the courts, perhaps for years, as various settlement attempts are made (mostly to fail). As for Sonora/Chihuahua, it is likely that they'd remain territories similar to Arizona and New Mexico well into the 1900s, so you'd have that mixed culture developing. While it may not be completely Anglo-fied, it would be much farther along than, say, Puerto Rico, simply due to being acquired far earlier and sharing a common market with the rest of the US, rather than being relatively isolated.
And, with the Americans thinking so little of the Mexicans at this point in time, it's not unlikely that they would attempt to use unsavory methods to relieve various natives of their possessions and deeds, some successfully, some unsuccessfully, which could result in a low grade conflict developing with land snatchers and hustlers fighting each other over property and land rights.
The foreign intervention in Mexico didn't start till 1861, but we are looking at a very different Mexico. The principle cause was reneging on debts to France, but in the TL the US would need to pay more for the lost territories ($25,000,000 was the highest authorised figure), and the political situation would be very different. To an extent, the French intervention was actually taking sides in a civil war, so I'd say it's far from clear that a different Mexico would face the same problem.
That would only be 10 million more, and I'm not sure how much the loss of the more developed territories, Chihuahua silver mines, taxation from Tamaulipas/Nuevo Leon/Coahuila, etc would come from that. It'd certainly eat into the difference, and there's no telling whether the Civil War will erupt or not (though I doubt these annexations would eliminate the underlying cause that lead up to it just as much as they don't alter the calculus leading to the American Civil War).
And, in the event of the American War occurring, the region would be a net drain on the Confederacy - either it provides a region where Union (or rather, anti-slaver factions) would be able to retreat and find shelter, which turns even more of Texas as a battleground (Texas would be in an uproar! The Rio Grandians would be coming to claim the Nueces as the border between them and Texas! Troops must be diverted to stop this), and there would be no more slavers there, relative, than there were in New Mexico historically.
Or, let's say, that Mexico
does intervene in an attempt to take back the territories - the Confederacy would then find themselves in a two front war, as their western claims (basically their OTL ones + this Southern California + Sonora/Chihuahua/Rio Grande) would be assaulted by Mexico, and they'd be in even more trouble than OTL. And, when that is all said and done, what better way to unite the North and South than to march together and expel the Mexicans from American territory? I don't see a way in which Mexico regains the territory, unless we have some Confederate/Mexican deal that somehow doesn't turn into backstabs... and a deal that would make an actual appreciable difference in the war, as the war will be won in the East as it is.