Assume the following:
- ACW happens as IOTL.
- Maximilian wins in Mexico.
- France wins the Franco-Prussian War.
That's a pretty big gap between the first two points and the third. The United States would not have spent 5-6 years doing nothing while a French puppet sits on the throne in Mexico, especially since they had the largest and most experienced army in the world at that time. If, in the years after the Civil War, the French persist in Mexico despite the United States' threats (which Napoleon III would have brain dead to do given the campaign's growing unpopularity at home), the United States would have intervened and toppled Maximilian in short order. Consider that, at the height of the French occupation of Mexico in 1863, the French Foreign Legion was less than half the size of the Army of the Potomac alone (39,000 to over 100,000, respectively). In total, the United States had over 600,000 well-equipped and battle-hardened troops at their disposal by the end of the war. France would have had no chance against such superior numbers, especially when growing tensions in Europe would have forced the French to withdraw troops from Mexico.
After the Emancipation Proclamation, the United States was immensely popular and would have been supported by virtually everyone in Europe outside of France - especially by the British, who I'm sure would have been very excited to see their ancient rivals getting thrashed in an overseas venture.