Mexico is ethnically/racially distinct from the US, and also religiously/culturally distinct. I think its possible to finesse one of those, but not both. So you probably need some kind of widespread protestant movement in Mexico. Hard to see how that would happen, but maybe something that sorta lined up with class/ethnic distinctions, i.e, the Indios are all protestants, or the more blanco/middle class types are protestants in reaction to some kind of militant Indio Catholic movement. A religious split like that could lead to lots of unrest and civil war and possibly lead to a large, significant faction *within* Mexico pressing for annexation. You probably also need cotton plantation agriculture to never become significant in the US to dampen down the slavery issue--earlier boll weevil?
It would help if there were also some kind of outside pressure on Mexico which would lead Mexican elites to favor US annexation. Maybe, ah, a Napoleonic expansionist Gran Colombia? Or a Spain that in the first half of the 19th pulls a proto-fascist economic and ideological retrenchment. Result: well-funded, competent, and thoroughly brutal efforts to take back Mexico. The US and the Mexican elites would be natural allies in this efforts, especially if the Spanish ideology was some kind of steroidal Catholicism that the Pope hitched his star to, making it easier for Mexican elites to go Protestant, or at least some kind of Mexican version of Gallicanism that Americans could accept as sufficiently protestant for getting-along purposes.
Another possibility would be a aggressive foreign power actually ruling the place for awhile and brutally imposing some kind of non-Catholic religious settlement. Not very successfully, but enough that when the US later helps the Mexicans drive the power out, there is a sufficient local Mexican *Protestant grouping for the US to feel it can work with.
Not that I have any problems with Catholics, but I think you have to have a Mexican Catholic-screw to make the challenge work.