Emperor Maximilian would probably have pursued some of the successful policies of OTL Mexican President
Porfirio Díaz who ruled Mexico essentially from 1876 until 1911.
Díaz said:
Díaz sought to attract foreign investment to Mexico to aid development of mining, agriculture, industry, and infrastructure. Political stability and the revision of laws, some dating to the colonial era, created a legal structure and an atmosphere where entrepreneurs felt secure in investing capital in Mexico. Railways, financed by foreign capital, transformed areas that were remote from markets into productive regions. The government mandate to survey land meant that secure title was established for investors.
I’m thinking, with the French withdrawal either by 1867 and most certainly after the 1870 debacle of the Franco-Prussian War, that Austria would step up in the place of France to establish firmer links with Mexico. Not only is Emperor Maximilian the brother of Austrian Emperor Franz Josef II but the romance of re-establishing the Austrian Habsburg connection with Mexico which existed in the 1500s would be appealing on several levels.
This time period would be before Austria had reconciled with, and allied with, Bismarck’s Germany. Austria could not hope for a successful war of revenge for her 1866 defeat against the Germans but she could recover some prestige and a new territory for investment and economic expansion ruled by a fellow Hapsburg. This outlet could also compensate Austria for not having a colonial empire of her own.
Empress Carlotta’s brother, Leopold II, was the King of the Belgians and was always seeking colonies, like his infamous Congo Free State. Maximilian and Carlotta might lease her brother some territory to develop.
And who knows but, if things had turned out differently, Maximilian might have emulated Leopold II’s later OTL acquisition of the Congo Free State by buying Alaska from the Russians as a present for Carlotta only to discover just how rich in mineral resources Alaska was which would also benefit Mexico.