Thing is that - as Jefferson famously and astutely noted - New Orleans is rather an important city. Whoever holds it can threaten the United States, it's important for trade... basically, the USA really wanted it. The Louisiana Purchase was actually a fluke, though. Jefferson was trying to get only the city of New Orleans, and free right of transit on the Mississippi (which the USA used to have, until Spain revoked the relevant treaty in 1798). So that could be a POD right there: Jefferson still wants New Orleans for the USA, Napoleon still grabs Louisiana off Spain and still wants to sell it, but Jefferson has more trouble getting Congress to agree to the Louisiana Purchase, since the USA already enjoys free right of transit on the Mississippi. Not many people wanted to buy all that distant hinterland. So say Jefferson eventually manages to buy only New Orleans. The rest stays vaguely French, until Napoleon is defeated, at which point the ownership of the territory becomes a point of pride for the mexican revolutionaries - and they eventually claim it for Mexico, agreeing to perpetually cede New rleans to the USA, and to make the Mississippi an international river with free transit for all.
The problem I see with that is that the USA had plenty of drives to expand west. Maybe not even actively, politically... but there were a lot of settlers just trekking west on their own. So it's rather hard to avoid Louisiana eventually being gobbled up by settlers. Maybe if Mexico saw that risk, and made it a policy to heavily settle Texas and the west bank of the Mississippi with Mexicans? Then the area would be more densely populated, harder to simply grab. If Mexican-American relations are also good in TTL, your desired outcome could come about.
Your scenario could be further aided by having Mexico agree to cede all claims north of the 42nd parallel to the United States. If the USA then dukes it out with Britain over Oregon as in OTL, the USA will get access to the Pacific anyway. This would both (somewhat) reduce the incentive to grab Mexican land, and ensure that the Missouri is no international border, per your request.