how about politically neutral "estado.....

How about the politically-neutral "estadounidense" for use in place of "Americano?" It is correct Spanish, and offends no one, although one never knows! It IS a bit of a mouthfull (just spit it out fast), but your Mexican friends, neighbours, cab drivers et al will be impressed. Win/win use of a second language.

You should have been a politician.

HOWEVER....isn't the full official name for Mexico "Estados Unidos Mexicanos"?
Therefore, wouldn't it be just as correct for a Mexican to say he is "estadounidense"?
Which should then prompt the question: "Estadounidense de donde?" Or, "Los estados unidos de qué"?
To which the answer, as much as Rob might cringe in protest, would have to be either "de America" or "de Mexico".

By the by, that infallible font of all wisdom, Wikipedia, says the following:

"The United States of America (USA), commonly known as the United States (U.S. or US) or simply America, is a country......"

The prosecution rests, your honour.