The whole point of all of this is that as the French develop closer ties with Brazil than OTL, it might become clearer to the French that Pedro II would not have a direct successor, given that his own sons died in early childhood and only a daughter (Isabel) survived well into adulthood. In other words, the Brazilian imperial house is solving its succession problems by adopting a foreign prince, which would presumably be approved by the Brazilian legislature, and without the need for French troops. Maximilian could relocate to Brazil in the early 1860s, become the Imperial Heir Apparent for about a decade, and then take the throne, making everybody happy.
Therefore, it's a much better fit than with Mexico; Brazil had genuine imperialist sentiment and a history of long-term imperialism of its own, whereas in Mexico, the House of Iturbide ruled only briefly and with lots of people resenting it. Meaning, the monarchy was an indigenous, continually preexisting choice. This also mean, technically, that Maximilian moving to Brazil would have been no violation of the Monroe Doctrine. Besides which, Brazil (unlike Mexico) was way farther away from the US, and the US didn't have a huge naval fleet to enforce the Monroe Doctrine as far away as South America. The US also didn't have so many economic interests in that corner of Latin America at the time, unlike in Middle America and the Caribbean.