Possibly with a defeat in the war of 1812. But it would have to be more than just a defeat. We'd have to change British war aims. IOTL, the British fought a primarily defensive campaign (outside of the assualt on New Orleans). We would need the British to conquer the US and have the United States revert back to colonial status. Or at best be forced to cede New England and the Northwest Territories to the UK. Then you can get a United States weak enough for Mexico to become the dominant power.
The big thing that is going to hold Mexico back is population. In 1803, Mexico's population was 6.8 million. Roughly equal to the United States. Maybe even higher. But by 1884, Mexico's population was only 10.4 million. The US had a population of over 50 million by 1880. Now, granted, some of that population came from Mexico losing the Southwest in 1848. But not enough to balance out the population.
Economics also kills Mexico. In 1820, Mexico had a population that was 68% of the US'. But their GDP was only 38% of the United States. In other words, their per capita GDP was only 57% of the United States.
The more I look at this, we need an insanely early POD to make this work. Something like Spain actually investing in Mexico and seeing it as an integral part of their empire and building Mexico as early as the 1700s, the way Britain did with the colonies. Otherwise, I just don't think it happens.
Admitedly, this could need a PoD earlier than 1821. I could think of a War of 1812 that goes bad for the US, perhaps even a defeat. Then add a successful Hartford Convention and New England leaves as well.
Keeping it post-1821, then perhaps UK-US relations take a dive somewhere, and the Oregon issue or that one that settled the Maine border turns into an actual war.