that might not be as big an issue as you'd think. First, the US tended to swamp native populations with it's own civilians and immigrants... the real question is if the land in the border states is attractive enough to draw southerners there. Second, while slavery was technically illegal, it's been noted by several historians that the peonage system in Mexico was scarcely better, and that wealthy Mexicans had slaves, although they were generally kidnapped Native American children used as house servants instead of the agricultural slaves used in the south. So, if you have a situation where southerners swarmed into the border states, slavery might not be such an issue there.
The problem you have there is, why swarm settled territory whose fertile land is taken? And who's going to swarm? It's difficult to overstate how empty of Mexicans New Mexico (incl. 5 modern states), California and Texas were: Less than 100k people between the three. Of course they were swamped. The border provinces had 6-7 times as many people in a 10th of the area. To swamp that population you require a movement of 700k settlers, which would be 3% of the US population, into populated territory with little available land. Meanwhile, in California...Gold!
And, for this to have the effect of creating a slave state, the vast majority of settlers must be southern. You are quite right about Mexico's social ills, but there is a long way from that to opening up slave markets. I just don't understand how it is conceivable to have these states as pro-southern slave-states within 12 years, it requires not just migration, but an exodus from the South, and the North to be fine with that.
BTW, in this scenario, whatever happens, the region rebels during the Civil War, probably with support from Mexico. Chaos ensues.