Here's a rough TL:
The Mexicans, due to a less bloody/destructive independence war, retain the Empire, and manage to win a fairly close victory in the Mexican-American War, retaking Texas, but recognizing the 1818 borders in the Rockies and Oregon. The US goes into a funk, which eventually results in a Civil War sometime in the late 1850s as the North forms a even larger proportion of the population than OTL. The North wins by 1860, with some covert Mexican aid for the CSA. This is discovered by the US, and tensions rise again with Mexico. War breaks out sometime in the late 1870s over a Mormon uprising (hey, they're at least white and English-speaking). The US wins this round in a close victory, creating an independent Deseret and annexing Texas north of the Nueces. The Mexicans fulminate over this, and overthrow the Emperor. After a long period of instability, the democratic government start a second, halfhearted war with the US sometime in the 1910s to shore up its power, which does little other than the US annexing Texas to the Rio Grande. The Mexicans then enter a civil war, and reinstate the Empire, which promises to return Mexico to the glory days of the 1840s. They make common cause with Gran Colombia, which has its own issues with Brazil over the Amazon Basin and influence in South America. On June 22nd, 1941, Mexican forces, with token commitments from Colombia and its allies in South America, cross the Rio Grande into history.
They manage great gains, as the US had stagnated militarily after the previous war and Mexico had furiously modernized and expanded its forces. They reach their high point in late 1942 around St. Louis and Vicksburg, while Seattle holds out heroically as a Leningrad-equivalent, and the Mormons starting a vicious partisan war against the Mexican occupiers. They get shattered before St. Louis, and their main forces get annihilated in a long retreat over the Great Plains under constant pursuit from avenging US armored forces. They briefly throw back the US advance as they cross the Rio Grande, and subject the US forces to horrific urban combat over San Francisco, but by the end of 1944 the US is besieging Mexico City itself, while US forces land in the Yucatan in cooperation with local rebels, and the Brazilians advance simultaneously on Caracas and Bogota from the Gunieas and Peru. The final peace treaty breaks up the Mexican Empire, leaving a core that is demilitarized and subject to indefinite occupation, while the US sets up an independent California, Rio Grande Republic, Central America, and Yucatan, and annexes large swathes of the Rockies and Panama. The world then settles into a three-way Cold War between the democratic US and its ally in Brazil, the Sydicalist Russian Federation, and a monarchist Empire of Germany.
That was longer than I expected.