It's always sunny in Mexico

In Chapter 13 Part 2 we come to some very sunny times for the Union. As many may know the French in 1861 intervened in Mexico in order to recoup certain debts claimed by the French government incurred by President Benito Juarez's government during Mexico's Reform War. Of course, the author makes a point of exaggerating the early Mexican success against the French (lest you think he's being inaccurate about the British alone):

Juarez had led the national government through the worst, including the victory at Puebla that sent the defeated French reeling back toward Veracruz.
The results of the battle had been a surprise to many; 6,500 French regulars, well-armed with M1857 percussion rifles and modern artillery, against 3,800 Mexicans, mostly irregulars, volunteers, and militia, armed with a motley selection of small arms and artillery, mostly cast-offs from the Napoleonic Wars, a half-century or more earlier. The French officers were professionals; their commander was Charles de Lorencez, a 47-year-old graduate of Saint-Cyr who had risen from third lieutenant to major general in the course of a 30-year career. The Mexicans were led by Ignacio Zaragoza, 33, a former seminarian who joined the Liberal cause as a volunteer officer in the 1850s, fighting against Santa Ana and the conservatives in the civil war of 1857-60.

What's curious is his claim the French retreated all the way to Veracruz. They did not, they retreated to (and fortified) Orizaba, allowing the French an inland base, and not cutting them off. Despite attempts to drive them off, the Mexicans proved unable to do so.

The author also seems to decline to mention the Mexicans actually had 12,000 troops in the area, and most of them weren't "irregulars, militia, or volunteers" as he portrays. Mind you, many of them were tasked with dealing with the 2,500 Conservative cavalry who were supporting the French in the area at the time.

In any event, the author wants to downplay as much as possible any inkling of success by the French in this period (lest he undermine his own premise a European power is incapable of fighting a war at trans-Atlantic distances) and proceeds instead to a matter of supreme convenience to the United States,

In the summer of 1862, after Puebla, the American minister, Thomas Corwin, had approached the Mexican government about a change in the American position of support, still largely tentative and mostly covert. After a lengthy series of discussions that took most of the summer and fall, the Mexicans had approved the commissioning of a United States Military Mission to Mexico; publicly, this was simply to take the form of observers, not unlike those currently accredited to the armies in Virginia, or those that had faced each other in the Crimea a decade earlier. The diplomatic niceties masked the reality that the two republics, each facing a foreign enemy and internal dissension (the Mexican state of Yucatan, whose people were largely Mayan with long tradition of resistance, was in rebellion), were looking for ways to maximize their respective strengths.

(long rambling exposition on how clever the Americans are and the experience of the men sent)

The likelihood of the tentative offers that Corwin and Shufeldt were prepared to actually make, however, would have belied what the alliance became; as it was, Corwin’s instructions were limited to offer essentially what had been discussed with the French in November. This amounted to an offer to pay Mexico’s debts to France, all 2.8 million pesos, in specie, and in return for a French withdrawal from Mexico. Settlement of Mexican debts by the Americans was, of course, offered only to the French; the British and Spanish claims were pointedly ignored. 

This in turn, would be the public offer; privately, the United States would pay France and Mexico to provide supplies – French powder and saltpeter across the Atlantic, Chilean and Peruvian saltpeter north in neutral ships to Mexican ports and sent north to the United States – from Guaymas north to Santa Fe.

As we already know, France was not interested in dealing with the Union on this matter. Furthermore, the sum of 2.8 million in specie is laughable compared to France's ever increasing demands from the Convention of London. Nor are we actually given a reason why the French would accept this offer other than authorial fiat and a heavy dose of handwaving the complex issues surrounding the French intervention historically.

All in all it really looks like a connivance on the authors' part to fix some of the problems he has discovered. This includes getting the French to drop their historic preference for the Confederacy, show an ahistorical aversion to Britain despite outright declaring their intention of pro-British neutrality, and going so far as to arrange to sell supplies to the Union versus the Confederacy. The other of course, is fixing the Union's gunpowder problem by opening a long and convoluted supply system through Mexico to import South American saltpeter to the North.

The author is seems, has realized the United States is far less than an autarky in this period and so contrives to ease their problems by means of handwavium.

So in order to seal the deal with Mexico:

What they asked for, of course, was something the French could not provide, something the Americans could, and something that would sever the Mexican conservatives from the French. What the Americans offered was retrocession of the Gadsden Purchase, the 1854 agreement where the United States agreed to pay Mexico $10 million for 29,670 square miles of land necessary for a southern transcontinental railroad. Given the purchase had been made during the Pierce and Santa Anna administrations, largely to benefit Southern interests, Lincoln made the offer, Juarez accepted, and the Treaty of Alliance was signed.

And so he retroactively concedes almost 30,000 square miles of territory to the Mexicans.

Whether this would be palatable to the United States as a whole (much less the precedent it sets) is an open question. One however can see that this could possibly be done were the Union desperate enough, but we can quibble about how effective it would be in actually securing the hoped for supplies.

What is certain, is that it will always be sunny in Mexico for the Union, and everyone will bend over backwards to help them.