WI: Napoleon III successful in Mexico, but loses against Prussia

What would happen if, somehow, Napoleon III successfully established the Second Mexican Empire, but still went to war with Prussia in 1870, and still lost, but avoided capture?
 
He might flee to Mexico, but I'm not sure that's a viable option. He'd still die a broken man soon after - after all, there's no chance he'd be able to usurp the throne from Maximilian.

The situation would stay the same even if he decides to fight on in the wake of Sedan. Sedan would be such a crushing loss that there would be demonstrations in Paris against the 'incompetency of the government', the republicans which had been gaining strength in the Assembly will make their move and overthrow the Empire, and Napoleon III would be hunted down as a fugitive.

Actually kind of like how Napoleon I was treated after his defeat at Waterloo.
 
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