So, we can come to the conclusion that the French in a hypothetical case of war would be put on the defensive pretty quickly by an uneasy alliance of US forces and Juarez forces, while the French navy tries to make short work of the US Navy.
In my opinion both sides have a limited window of opportunity before being forced to give in:
- The Americans are still recovering from the Civil War, so I expect that the intervention is unpopular with the common man. The French navy might not be capable of fully blockading the US, but they will certainly harm trade and commandeer/destroy American vessels. There will be tension between the US expeditionary force and the Mexican resistance.
- The French have no chance of outfighting the US in the America's, but they might break America's will to bleed over Mexico. However, they will be watched by the other Great Powers. Britain was never too enthousiastic about the French adventure and might force France to quit the war or face the Royal Navy, especially if British trade with the US would be blocked. At the same time Bismarck will certainly try to profit from this French distraction and the troops in Mexico might be needed in Europe soon.
Who will outlast who? Thoughts?
It all depends on your own personal prejudices I suppose.
If you think the entire American Civil War was worthless for military experience, that the US will give up quickly on the war while the French employ Skippy the Alien Space Bat to deploy their entire army and navy with lightening speed while keeping all of Europe out of the war, then yes, French total victory.
If you think the Union emerged from the ACW with the most technologically advanced, largest, and battle hardened army in the world? With men on the Texas border facing French colonial troops spread out over the whole of Mexico? With the French desperately trying to hold on against a now well-supplied and well-armed Juarista army ready to co-operate with the Americans (remember, 75% of the US Army that fought in the Mexican War were Southerners)? With a Prussian Army and strong government in Berlin looking to finish up uniting Germany?(1) With the British fed up with Napoleon's pointless adventurism, and the USA enforcing a Monroe Doctrine that London had quietly supported all along? Then yes, American total victory.
Neither side will see their army in the other's capital.
1) Interesting to see how the Austro-Prussian War would play out in this world.