A delayed Franco-Prussian war, or Max gathering support from another benefactor, would have been enough to keep Max in command.
I doubt good old Max would still be alive in 1914, the obvious successor wouldn't be of his dynasty but his adopted heir, the grandson of the previous Mexican Emperor. I suppose its possible Max could have a kid though.
Why? It is not impossible, he would be 82 years old. Sure that is a bit OLD, but Diaz was two years older and still alive (although retired in Paris by the time). Maybe Max dies in the early 1900s, and the Agustin or Salvador (his adopted sons) take over.
However, Mexico would be ripe for revolution at the time. No matter, how liberal Max is, the truth is the Mexico would be in a situation very similar to how it was under Porfirio.
Meaning, a large powerful elite made mostly of the old Mexican elite and recent European and American immigration (yes a few Americans immigrated to Mexico at this time and made it been notably the Creel and Sanborns families) controls most of Mexico, there is a huge underclass of ignored farmers, droughts in 1907 and 1910 and the US finding better and cheaper sugar investments in Cuba (two things I don't see changing) mean the price of corn and sugar have skyrocketed.
As I said, under any administration of the kind, Mexico is ripe for revolution.
The only way to avoid this, is if the Empire had diversified economically (unlike OTL), some early railroads wouldn't help, further investment in other crops like cotton or coffee, as well as port infrastructure. Whether the Empire under Max could have achieved this when Porfirio, who did a lot for Mexico couldn't, is the big question.
Otherwise the Empire would have to adopt populist policies, to control the price of food. Whether the Empire is willing to do this would be hard to know, and the long term effects of such policies are quite harmful.
As any populist government in this position one of the policies that Mexico could adopt would be increased militarization, and nationalization. The promise of recouperatinging territory might be appealing in this case. However, Mexico is likely to just join the war on the side of its sponsor (France if it is still interested), and if France and the US happen to be aligned then it might not be happening.
Also if Mexico does invade, the war would end at a stalemate at best or Mexico would get its ass kicked quite badly (again).
