A complex mix of political and social factors kept the US from going to war with France and invading Mexico. To get our pretend war, we have to assume enough catalyzing events to increase the resolve of the French, the resources available to the Imperial side, and the readiness of the US to enter a war so soon after finishing the bloodiest conflict ever seen in the Americas.
Europe is quieter
Napoleon III had to limit French involvement in Mexico because he also had military commitments in Rome and Africa. In addition, Bismarck was a loose cannon on the deck of Europe, and Napoleon was extremely wary of him. Prussia's lightning defeat of Austria in 1866 exacerbated his fears.
The campaign assumes Napoleon finds another way to mollify French Catholics and pulls out the 8,000 man garrison protecting Rome from the Italian unification movement. This gives France a whole extra division to send to Mexico.
Napoleon is more persuasive
Napoleon tried for many years to get the French Parliament to pay for a larger army. He was constantly worried about a big war with another European power (likely Prussia, the United Kingdom, or Italy) and wanted to have the biggest and best prepared army in Europe. In real life he was unable to get funding for the Garde Mobile until the disastrous Austro-Prussian War let him say "I told you so" to his own government, and by then French troops were coming home from Mexico.
The campaign assumes Napoleon was more convincing and persuaded his own government to fund a larger French Army. This in turn allows him to commit more forces to Mexico.
Prussia is not so tough, after all
Napoleon III considered Prussia the greatest threat to his growing French empire, and he focused a tremendous amount of his efforts on competing with them. His rival Bismarck eventually exploited this belligerent attitude and goaded the French into declaring war in 1870. However, Bismarck's primary focus was always German unification, and he carefully avoided diplomatic contests and wars that didn't assist this aim. Prussia might very well not have fought any wars with the French if Napoleon had focused on other goals and left alone Bismarck's project to create Gross Deutschland.
The campaign assumes that Napoleon III decided to take a purely defensive and diplomatic approach toward Prussia and concentrate his military resources on easier avenues of expansion. This leaves him free to commit a much greater proportion of his huge military to the Mexican endeavor.
However, the campaign also assumes that Napoleon's very real problem of financing the Mexican intervention persists (Mexico was not a terribly rich prize), and he would prefer to keep the operation from growing out of hand. The French commitment in Mexico at the start of the campaign is approximately what was really there in history. The "extra" forces are all available as reinforcements, to be committed only in the event that the US is foolish enough to actually invade. The USA was weary of war and a fair weather friend of Mexico, so the campaign assumes Napoleon is counting on US isolationism, a strong French position in Mexico, and the threat of another long war to defuse US belligerence.
Juaréz is captured
In 1865, just after Lee's army surrendered at Appomattox, Juaréz and his government-on-the-run found themselves cornered in the north against the Rio Grande. Marshal Bezaine, the commander in chief of all French forces in Mexico, had boldly been chasing Republican detachments all the way into Texas, and even accidentally fired on US troops and barracks a few times. Napoleon backed down about as far as he could shortly thereafter, and a war with the US was averted.
This campaign posits a minor event in mid-1865 with major political ramifications: Juaréz is captured, on the wrong side of the border, and US cavalry troopers are wounded and killed in a minor clash while trying to enforce US sovereignty on Texas soil. Juaréz is marched back to Mexico City to face a show trial and firing squad, effectively decapitating the Republican cause. Sensing a cementing of the French position in Mexico, US politicians have all the cassus belli they need, so they decide to cease rattling their sabers and draw them. The invasion can begin.
Jumping off
It took until summer of 1865 to gather enough US troops in and near Texas to threaten an invasion of Mexico, and in truth it would have taken more months to prepare the logistics for real marching and shooting. Mexico is a big country with rugged terrain and no major waterways penetrating the interior, and in the 1860s it also had limited resources and no rail network. The US Army had discovered in the 1840s how hard it was to make the 700 mile march to Mexico City against opposition, and Sheridan's 1865 Army of Occupation was much larger than the armies of the Mexican American War, which would complicate logistics further. A real war in Mexico would have had to be waged partly along the coast and partly along the interior line from Monterrey to Mexico City, with supplies being brought as far as possible by water and then hauled overland by mule and wagon trains. The only rail line in all of Mexico barely reached halfway from Vera Cruz to Mexico City.
Since both sides would have been dependent on naval forces, the weather becomes a major concern. The Caribbean hurricane season lasts from June through November, and naval operations in the Gulf and Caribbean were normally reduced or curtailed during this time. Only very small forces were operated more than a half a day from a safe harbor, and large fleets were typically sailed out of the region altogether to avoid massive losses to angry storms.
The campaign assumes a start date of December, 1865. This allows time for the US Army and Navy to stage the forces and supplies for a campaign, prepare a coordinated plan of action, and avoid naval operations in the hurricane season.
The French would have observed these preparations, and had some time to bolster their defenses.
The actual plans and preparations are left to the campaign players to decide.