It's interesting, which is certainly why Max made the offer, but:
I know this is highly improbable given that Juarez was the duly elected President of Mexico. At any rate, I had an idea where Carlotta dies en route to Mexico and Maximilian decides to marry a Mexican noblewoman in order to try to win over the Mexican people. Maximilian makes his same offer to Juarez to be his Prime Minister and have a liberal Mexican Empire. But, this time, Juarez decides to accept Maximilian's offer and be his Prime Minister. How do you see this playing out and can you think of any potential Mexican brides for Maximilian?
It's interesting, which is certainly why Max made the offer, but between 1857-60 the Mexican Liberals and Mexican Conservatives fought a civil war that was the bloodiest in the country's history as an independent nation to that point; the Liberals won, which is why Juarez was the first legitimately elected civilian president in the country's history.
Then, a year later (1861), the Conservatives come back after intriguing with the French, and there's another bloody war.
Then, Max is offered the throne by the Conservatives, propped up by French bayonets, and Max comes ashore in 1864.
It requires a Liberal cause, and a Liberal leader, who is neither of those things to even come close; the equivalent would be if Lincoln survives the Civil War, which ends in a US victory over the rebels in 1865, and then in 1866 the British attack with a foreign prince as the figurehead and offer Lincoln the premiership.
It's lunacy, basically.
Juarez was, very much, Mexico's Lincoln; he's not going to turn traitor in what is obviously a temporary occupation.
Best,