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I think the US would attempt to destroy the Mexican government. If it came to it, yes they would invade, although first they would simply try to install a friendlier government through a coup of some sort (it has been US policy to oppose any such actions since Monroe doctrine). As for the feelings of the US public, they can be swayed to war, as history shows.

JFK almost went to war with the Soviet Union when they tried to station long range missiles in Cuba.

Not only would the US invade Mexico under this scenario, they would shock-and-awe roll it over.

https://en.wikipedia.org/wiki/Monroe_Doctrine

The US did preemptively invade Panama in the past, at the first sign of General Noriega's shift toward the Soviet bloc, after his soliciting and receiving military aid from Cuba, Nicaragua, and Libya.