OOC:
Lincoln's kicking the French out of Mexico was holy writ. He had been sending letters pleading with President Benito Juarez to keep on fighting the Imperial French at all costs. "God willing, our own civil war will soon be over. When it is, I promise to do all in my power to aid your people in gaining their freedom"-
Abraham Lincoln, in a letter to Benito Juarez following the fall of Mexico City.*
It was Mexico's victory over the French at the Battle of Cinco de Mayo in 1861 that delayed the Mexican Conquest for a full year. By the time the Imperial French reached the Rio Grande in the Summer of 1863, Gettysburg and Vicksburg had already happened, and Nappy III was forced to be much more circumspect regarding his relations with the Confederacy.
The region of the Confederacy known as the Transmississippi (Texas, the Indian Territory, Arkansas, and western Louisiana) was better fed, armed, and clothed thanks to uninterrupted supply lines from French occupied Mexico. One of the reasons the region was basically bypassed until the end was that the ability of the Texans to resist was so strong. A good reason for Lincoln to look for payback. The disastrous Red River Campaign is another.
Immediately upon the surrender of all Confederate forces by now General-In-Chief Joseph E. Johnston (including Kirby Smith's command in Texas), General Grant ordered SHeridan to take four Union Corps straight to the Rio Grande. That's twice as many troops as the French had in all of Mexico. The Union was totally mobilized for war by that point, with the largest active modern army in the world. The French took off within the year. And Maximillian wound up in front of a firing squad.
Even had the US taken the ASB decision to leave an Austrian Emperor on a phantom Mexican throne, the French could not have stayed. With the Franco-Prussian War, the need of French troops in Mexico to get home to fight, the fall of Nappy III, the Commune, the surrender to the Prussians, the Rise of the Third Republic, and the ruinous indemnities the French had to pay the Prussian/Germans, Maximillian would be left fighting a heavily reinforced and rearmed (US ARMY surplus

) Mexican Republican Army with only a handful of Imperial Mexican brigades of very dubious loyalty.
With the Third Republic obsessed on Revanche, no way in HELL does a democracy like France go to war with a democracy like the US. The French are going to go to war with the US after they've lost Alsace-Lorraine to Germany and have a powerful, United German Empire, with a huge army on their borders, pointed at Paris!?
Then there's the whole "We just spent all those francs building, shipping, and assembling the Statue of Liberty for the Americans as a friendly gift between democracies, and now we're supposed to throw away all that good will on a few piles of sand?"
