He may refuse to surrender the province, but he wouldn't dream of declaring independence from Europe; the last riot that burned Mexico City down was less than a generation ago, he's dependent for a great many functions on the Church in Mexico which is in turn dependent on a European monarch (no Viceroy of New Spain would be pleased with the results if Rome got to appoint its archbishop and bishops). He'd declare that New Spain was the property of the House of Habsburg rather than the Crown of Spain (meaning they'll try and rename the place?) and pledge himself to Vienna. And Britain would happily accept that state of affairs, which makes it likely to stick.
The question becomes whether this triggers a cascade among the other three viceroys? Do they declare themselves loyal to the nation of Spain or the monarchs in their person? Again, Britain would be most pleased if two went German and two stayed Spanish....