Actually, Austria-Hungary hadn't much designs in Americas. As for Napoléon III, his main objective in Mexico was to set up a stable regime (by contrast to the serious instability of the republican regime), a catholic state that would be able to check US power and influence in the region. Now, even the restoration of an Empire in Mexico was an accident at this stage; Napoléon III wanted to wait for a proper referendum to be held before doing such a thing (I hold this as coherent with the character since one of the defining points of his doctrine was popular sovereignty such as exemplified by his exigence of a referendum to confirm annexation of Nice and Savoy), but before it could happen, General Forrey and a group of Mexican conservative notables (I don't know whether the former or the latter initiated it) proclaimed the Empire, presenting Napoléon with a fait accompli.
Now, the main vector of penetration in America for France was through trade and finance.
But I imagine there is some opportunities:
- Panama. Some times ago, I outlined a scenario where Gustave Eiffel, taking over the canal project from its beginning, and by imposing his plan of a canal with locks avoids years of work and money being lost (though without the medical advances made by Americans, it would still be costly in term of human losses). In this perspective, France, and not the United States, would be in good position in the the 1880s to turn Panama into a banana republic (though that would require separating it from Colombia, but that's what the US did).
- Araucania and Patagaunia. Provided a pod involving either Chilean conquest being delayed or King Antoine being more successful and evading capture, you could end up with the cone as a client of France (for military protection).
Anyway, if you're speaking of the Franco-Austrian alliance in Europe, it's almost guaranteed if you have the Second Empire surviving, due to a need of checking Prussian power.