In my opinion the best chance for a strong Mexico is a near-absolute conservative monarchy under the House of Iturbide. This requires a POD of around 1822.
Americans would have unrestricted immigration to Texas, but if the monarchy was absolutist, I can't see them liking it unless they have autonomy. If they do, eventually their children begin to see themselves more as Mexicans than as Americans.
The new Mexican government refuses to recognize the treaty of Adams Onis, and with the added strength of Americo-Mexicans, likely helped by an earlier discovery of gold in Alta California, they go to war with the US over the Sabine Free State (both territories claim the other is the aggressor in what essentially is disputed territory, not too different from the start of the OTL Mex-Am War), and Mexico ends up taking Louisiana (particularly New Orleans), and the former West and East Florida.
Mexico also still holds Central America at this point, and I'd expect them to intervene in Cuba at the first sign of any anti-Spanish sentiment.
This could, I think, end up with an American-Spanish alliance against the Mexican Empire.