Uh...
The French were not about to get into a serious war in North America. The forces they did send were fortunate to hold off the Mexican irregulars for as long as they did. Benito Juarez proved to be a much more capable and tenacious opponent than anyone in Paris expected. They did ignore the U.S. during the ACW, but they started paying very close attention to the U.S. which suddenly had a couple million combat veterans, a lot of them between employment opportunities. The French puppet government failed to, among other things, interfere with the transfer of 30,000 stands of arms directly from the U.S. Arsenal at Baton Rouge to Juarez's forces (not that it would have been a good idea to try, Phil Sheridan was a very skilled combat commander).
I would also point out that most of the "Mexican Territory" in question was considered to be part of the CSA by the Confederacy (there were a number of little known but sharp engagements between Federal and Confederate forces in the Arizona and New Mexico Territories), with the Lion's share being the State of Texas. The Confederacy actually offered, during the Hampton Roads Conference to make common cause with the U.S. to evict the French.
It is also useful to consider that the French effort did not take Mexico City until June of 1863. Less than a month later the Confederate's last actual, albeit slender, chance for victory died on a hot day in Pennsylvania.