I think in a scenario with a very stable Mexico ~1820, you'd end up with an early period where Mexico and the USA are direct competitors/rough equals geopolitically early on but skewed in America's favor pound for pound.
The USA would have several major advantages such as better geography for a unified state, better topography for agriculture, a more developed economy, a large merchant marine, high literacy rates, and nuts birth rates. Mexico would have immediate access to both the Atlantic and Pacific Oceans, stronger soft power in the rest of Latin America and the Phillippines, be more appealing to a wider segment of OTL's 1800s historic immigrant populations due to being Catholic, and a slight lead in the settlement of the West due to their very light presence in California and Texas.
America's historic demographic ballooning would help them settle the Mississippi at the same rate as OTL early on, but Mexico would be far more competitive than OTL too due to their less restrictive immigration policies(just be a Catholic 4Head). The USA constantly flirted with nativism and anti-Catholic sentiment, which I imagine would be much worse in this scenario if Mexico is viewed as a competitor or even an enemy.
I'm sourcing Wikipedia here for this entire paragraph, but Irish immigration to the US started to ramp up from 1820 onwards reaching 195k in 1820, 663k by 1840, and 1.6 million by 1850. Irish immigration from 1820 to 1845 numbered ~1 million. Pretty clear that there was non-insignificant Catholic immigration going on here as a percentage of the immigrant Irish during this period. Then when you get to the Potato Famine, 90% of immigrants were Catholic, and ~1.5m-2m people left Ireland by the end of the famine.
I'm obviously imagining a scenario here, but a more stable Mexico is a more prosperous Mexico. You'll probably have greater trade between the UK and Mexico due to this as well as greater Irish immigration to Mexico long before 1846. When the potato famine comes around, narratives and trends may have shifted enough that it's now Mexico, not the USA that is the primary target for Irish immigration. Considering OTL Mexico's deal to settle Texas with Americans, with greater immigration already in place you might see an already very Irish Texas become the primary destination for Irish immigrants from the famine. Extremely Irish Texas incoming?
This was a little side speculation to highlight where there's a lot of potential variances. Looking at today's demographics, 1 in 4 German Americans today identify as Catholic, 1 in 3 Irish Americans today are Catholic. And that's before we get to the next runner-ups, English(no data on the number of English American Catholics), non-specific Americans, Italian, Polish, and French Americans all of which are going to have significant to supermajority Catholic numbers. A stable Mexico can serve as a counterweight to the USA and draw in a majority of these Catholics, especially the ones that don't come from an English-adjacent culture like the Irish, ex. Poles or Italians as well as siphon immigrants that OTL went to other countries in the Americas outside of the USA and Mexico such as Brazil, Argentina, and Colombia due to being the poster child for Catholic prosperity in the New World, control of California and its gold rush, etc. to give Mexico its own hype reel for immigration.
Stability in Mexico allows for a more organized settlement of El Norte, and I expect O'Donoju living could help play a part in broaching the Irish-Mexican immigration pipeline. Austin may still have his American colony in Texas, but he'll have to cohabitate with Mexicans, Irishmen, etc. being encouraged to settle at the same time which may blunt the settlers' aspirations in Texas to work around the Mexican government and introduce ex. slavery. A Fredonian Revolt analog would accelerate this and Mexico may actually be able to make good on blocking further American immigration to Texas or enforcing a quota as a compromise. Additional tensions could arise here since, until 1828, Mexico didn't recognize the Adams-Onis treaty and still claimed Oregon Country. This could lead to a much earlier Mexican-American War, and that's likely to play to Mexico's favor if anything as American ability to project power into the North American interior is as gimped for the USA as it is for Mexico, whose ability would be better than OTL but still absolutely terrible.
From there, who can say really? The USA and Mexico's ability to project power into Texas is limited due to how far it is from their centers of power but I'd say Mexico has a slight advantage due to their active buildup in Texas compared to the USA's more passive approach. The war could go either way there, though I'd anticipate American naval dominance in the Caribbean hampering what on paper would look like a slightly favored matchup for Mexico. But if Adams-Onis being unrecognized is still up in the air and tensions took time to build, then you've got the reverse is quite likely in the Pacific; Mexican naval dominance wrecking American presence in the region.
So the war ends up being very lackluster for both sides with the Mexicans losing Texas east of the Brazos and north of the Canadian River until it reaches the Continental Divide while gaining land in the Oregon Country; the border is defined by the Snake and Columbia River instead of the 42nd parallel, leaving all lands south to Mexico. Mexico ends up reviving the Spanish claim to the Pacific Northwest as a result, though limits itself to what it gained at American expense. Fort Astoria and Fort Vancouver being in British hands in lands claimed by the Mexicans raise eyebrows in London but Mexico manages to not fumble the entire thing and calls for a summit to settle the Oregon Country once and for all with America as the weakest party at the table. The end result is something like this:
* Mexico ends up with their borders as written in the peace between the USA and Mexico. All of Oregon, a chunk of southeast Washington, and a sliver of Idaho that's south of the Snake River are their gains
* The American claims in Oregon Country are defined by the Mexican claims to the south, and the 48th parallel up until it reaches the Columbia River, at which point it follows the river until it reaches the 49th parallel and follows the rest of the British-American border east
* The British border is all lands north of the American border(48th parallel until it reaches the Columbia River)
The USA wins in the primary theater of war, Mexico in the secondary, and a bitter rivalry is born. Mexico and the USA spend the next century competing to populate their claims in the interior of North America and for influence in the New World and the Pacific. The USA leverages its booming birth rates and the immigration of Protestants from Europe, having hardened its anti-Catholic stances after their loss of most of the Oregon Country and continued rivalry with Mexico. Mexico ends up receiving the majority of Catholic immigration that OTL went to the USA. Aside from a large number of Irish Catholics, Mexico receives many Spaniards, Italians, Catholic Germans, and Poles in this period and experiences decent demographic growth of its own due to its stability and growing economy. The USA's also likely to have stronger internal discontent than OTL; northerners blaming the southerners for the loss of most of Oregon Country for a small strip of slave land, and vice versa viewing the northern states as perfidious allies that could have done more to secure a greater victory over Mexico.
Choose your poison from here; fracture America, have the USA and Mexico keep relative parity, have one gain a late advantage over the other in ATL-World Wars, etc. Though I imagine that Mexico's probably more successful in both the New World and the Pacific than the USA when it comes to soft power due to their shared language and religion with the rest of Latin America and their existing ties with the Phillippines, while the USA whether its a rump North or not hard claps the Mexicans in industrial development.
