The more I think about it, the more I like the Mexican Pacific expansion idea. After all, what was Max best known for before he was Viceroy of Lombardy-Venetia? Being CinC of the Austrian Navy, which he modernized considerably. The man loved the sea, and promoted research voyages that gained fame in the day.Not sending Miguel Miramón to Berlin is another POD that could be useful, as he could become a vital piece building the imperial army. His military talent could also be a great boost to the campaign against the Republicans.
I agree with others that retaking Central America is mostly a pipe dream. However, Mexico could pull the Central American states closer to its orbit through other means without outright annexation. Better relations with GB could lead to the purchase of Belize or at least a border adjustment favorable to Mexico, something that, if I recall correctly, jycee did in one of their timelines.
Speaking about Pacific expansion, Mexican Hawaii, anyone? And perhaps restarting something akin to the Manila galleon.
I'll ponder more about the scenario and come back with more input. A thriving, surviving Second Mexican Empires is one of my favorite ATL scenarios for Mexico.
Edit: shout out to @KingSweden24 and his wonderful timeline "Cinco de Mayo" which explores this precise scenario.
Once the political situation in Mexico was stabilized, building up a strong naval and merchant fleet could become a matter of national pride, and promote unity between the factions (maybe).
A Mexican Hawaii would probably preserve the royal dynasty there. Hawaii was internationally recognized as an independent state, and its monarchy was evolving under a very European model. Becoming a protectorate, vassal state even, of the Empire would be more palatable than falling under the sway of a firmly anti-monarchist nation like the US (and its various corporate interests)...
And, speaking of Manila.... the Philippines were formerly de jure administered from New Spain. Max could take the opportunity of political instability in Spain (1st Spanish Republic, or any number of other events - just pick one, there were plenty to choose from ) and launch a "temporary occupation of the Philippines until order returns to the Spanish Realm"... which of course could forseeably become permanent
This would have quite interesting implications for Pacific realpolitik in the 20th Century