Maximilian knew he was going to die childless (either because he or his wife was steryle) and named the grandson of Iturbide as his heir after 'adopting' him.
Interesting, and I didn't know that until today. However, here's what Wikipedia says:
As Maximilian and Carlota had no children, they adopted
Agustín de Iturbide y Green and his cousin
Salvador de Iturbide y de Marzán, both grandsons of
Agustín de Iturbide, who had briefly reigned as Emperor of Mexico in the 1820s. They gave young Agustín the title of "His Highness, the Prince of Iturbide" and intended to groom him as heir to the throne. However, he never intended to give the crown to the Iturbides because he considered that were not of royal blood.
[24] it was all a charade directed to his brother
Archduke Karl Ludwig of Austria, as he explained himself: either Karl gave him one of his sons as an heir, or he would give everything to the Iturbide children.
[24]
So, it looks like the Iturbide claim can disappear, and then Karl Ludwig's line's claims can end up in Franz Josef's hands eventually, either by cession or by fortuitously-timed deaths of heirs.
I expected there would be other objections based on other country's attitudes. True enough, but the situation isn't completely hopeless. Briefly, France was already an ally, and pre-Bismarck Prussia would not be an immediate threat. Later, in fact, Bismarck was perfectly happy for Austria to expand anywhere but Germany. Spain and GB might have objected, also, but each also had an incentive to support any country willing to challenge the Monroe Doctrine.
That leaves the US, and there's no question it would be hostile. All the other independent countries in the Western hemisphere would be, too. However, you have to wonder whether the US would be ready engage in a war with Austria and its allies immediately after the Civil War.