If the French didn't back fold and the US did march 50,000 troops into Mexico, I don't imagine the French or Mexico winning. However, there is no real win in war, it all a matter of how much you lose in comparison to the other guy. One thing the French do have is modern equipment and a luke-warm populace.
In theory Mexico could figure an army the same size as the US, but that would take a much more liberal Maximilian. The POD needs to be a bit further back for that to happen. Even if this force was raised, it would have supreme difficulties in supplying it with guns, let along the sort of cannon needed to match the might of the US in a pitched battle.
If I were the French I would attempt to have this force rallied and preparations for a, this is the important part, long and drawn out campaign. For the US to achieve the sort of swift victory needed, it will need to move along the coast. Sweeping down from Texas and then heading for Mexico City. As the French it would be my objective make sure this doesn't happen. In the build up to the war I would fund and supply small groups of Mexican Soldiers, or better yet bandits, to lie in wait for the US Army to cross the border. I would leave these groups under the command of a reliable French Commander, to harass the US supply lines and to launch raids into Texas, as far as Huston is possible.
The majority of my force I would hold at the Tampico and Rioverde provinces. I would improve the defenses of the towns and settlements as much as possible. Fortifying the towns will force the US force to either take them by force, or face the threat of attack to their supply chain or rear flank. Even with the US Army's size, better equipment and experienced soldiers, would probably think twice before engaging the opposition at fortified positions. The US General would have to make the choice to months of cannon bombardment to weaken the defenses, forcing the battle, or attempting to circumnavigate the defenses and go via the mountains.
If the US choose to attack it would probably win the battle, but the loss of life would be a huge hit to the moral at home that would have thought this a quick and easy war. As the French this would be the time to call the best of my soldiers to the fray. Use what's left of the Mexican, French armies to hold Mexico City and to attack the supply lines. Beyond that it will be up to the US to decide who wins the war. As the US you could force an attack of Mexico City and again probably win, or wait months for the defenses to be weak enough. If it's the first you win the war, but with an appalling loss of life. Otherwise the support back home for the war will probably be dwindling rapidly and I would be surprised if the President faces calls to end the war.
If the US choose to wait, the support for the war will probably dwindle or the same. That quick victory will be forgotten as news of Mexican raids into the US quickly replace that dream.
If the US does choose the mountain route they will have given the French everything they could have wanted. A longer, harder route to Mexico City, with more places to attack and harass the US while the defenders fall back and the constant fear of being circled behind. The US Army would have to go at a much slower pace, leave garrisons in every town the faced, lest they harass the supply lines and devote troops to hunting down raiders and protecting supplies. As the French I would lie in wait outside Mexico City, let them get that far and then attack. The French would have a good chance of winning, and would end up harassing the US all the way back to the border. Of course this would not happen the next time the US invaded. The next US army would not make the same mistakes and would press onto Mexico along the shortest route.
In all these scenarios it depends of US public support for the war. If it's high they will win, if it is allowed to be drawn out then it will dwindle. The US is still fresh from the Civil War and they will not stomach a drawn out campaign to remove a leader of another country that isn't a serious threat.