Napoleon refuses to pull out after the Civil War ends, U.S. intervenes to topple Max?
Then Napoleon would be facing even larger troubles back home. He pulled out to concentrate on Prussia (which still didn't end well for him).
If NapIII decides to stay and help out Max then things back home will likely go worse for him. Max looses France support, Max is still overthrown, by the liberals (perhaps a few months later though), and things continue more or less the same for Mexico (though worse for France).
I think the best way to do this, is to have the conservatives react sooner against Comonfort and the 1857 constitution. Quick background for those who need Mexico enacted a very liberal constitution in '57, Igancio Comonfort was a "moderate liberal" who was elected President, everyone was unhappy after a few months in power Comonfort threw a coup against himself in hopes of re-drafting a new constitution. This led to Juarez assuming power, which led to the Civil War, that led to the Empire, that led to Juarez assuming power in the end once again.
Have the conservatives preempt Comonfort and assume power in '58, and kill off a few liberals. Once this stabilizes you have the conservative, military minded, centralist government you need (skipping over Max) to stupidly declare war against the US for any excuse.