I'm suggesting that he was a CINO and proudly advertised the fact, and was therefore unfit for office. CINO-ism is fundamentally dishonest.
I believe that neglect of civic responsibilities at the behest of a foreign sovereign
To me, as a Catholic, that's a silly argument. If the Pope were an Italian citizen, would your argument be different? If the Pope were an American citizen, would your argument be different?
What if we were speaking of the President of some hypothetical foreign country, who happens to be a Mormon? His spiritual leader is then (more than likely) an American. Would the same argument apply? Would you suggest that a Mormon would be unfit to be President of, e.g., Argentina, simply because he owes relgious allegiance to a church headquartered in a foreign country?
To a Catholic, the Pope is a spiritual leader. The fact that he is temporally the chief-of-state of a "nation" the size of a large city park is irrelevant. His spiritual authority would be the same if he were legally a citizen of Zambia.
If you think that people in government ought to ignore their spiritual leaders just because those leaders are citizens of a different country, I think that's silly and provincial.
I don't think your argument rebuts mine at all: the Pope isn't an Italian citizen, but rather a temporal sovereign as well as a spiritual leader. Furthermore, he is a temporal leader whose state has a 1500 year history of claiming temporal power over other sovereigns and which has never fully renounced its claims to temporal power. So, it is the very status of the Pope as a temporal sovereign that creates a part of the problem. You can't wish that aspect of the problem away without abandoning the notion of Papal sovereignty. (And, there is a lot of history about relations with the Vatican by temporal states as well).
As to your Mormon example, if the First President of the LDS Church claimed the right to order a Morman President of (say) Argentina to act or refrain from acting, in violation of the law of Argentine, then I would argue against electing a Mormon as president of (say) Argentina. And, if elected, if the Mormon president of (say) Argentina were not prepared to put the law of Argentina first, then I would argue him to be unfit for office.