If we assume that the Church’s authority to legislae morality is only binding on Catholics, then we become the catholic church, with lower case “c”. The Church’s place in the world is for all people, not just those who physically belong to her. This takes us back to the dogma of no salvation outside the Church.
The Church has the duty to save souls, inside her physical body and outside. She also has that power given to her by Christ. This is undeniable. Therefore, even those who are not Catholic are morally bound by the moral teachings revealed through the Church. That’s how salvation reaches them. Christ uses these means to save. But if we assume that the Church’s moral revelation is only for those within her physical boundaries, then we are returning to the Old Law that the Apostles told us was abrogated. Just as salvation was not just for the Jews, it is not just for Catholics either.
When you have two codes of law, side by side, one is revealed by God and the other is created by man, and they are in conflict, you have the mandate from Christ himself to preach the Gospel. This means that you have a mandate to censure the man-made law that is in conflict with the revealed law, because that man-made law stands between man and salvation.
The other risk that you run, if you do not insist in showing how the man-made law is a lie, you run the risk of sinning against justice. In justice, every human being is entitled to be told the truth. In justice, God does have a place in the public sector. The state does not have the right to deny him this.
God does have a place in the public arena and the public arena must never ignore revelation or revealed truth. It is the mission and moral duty of every Catholic to remind the public arena of this reality and this obligation. You cannot restrict moral truth to the members of certain faiths and let everyone else go their way. To do so, you are denying God his rightful place in society and you are allowing society to believe that there can be two codes of conduct that are mutually exclusive. To allow society to believe this is to allow a lie.
You also mentioned that this has not done well in the USA and I agree with you. This is what some popes have called the American Heresy. I don’t believe it’s just American, but I guess that it can be called that, because we were the first nation to implement the notion that freedom of religion meant freedom from moral laws. But this thinking is wrong. The moral law has been revealed through the human conscience and through natural law. Man is bound to it. American Catholics have a moral duty to impress moral revelation upon the its elected officials and upon their fellow citizens. What we cannot do is proselytize. That is forbidden by the Church. But we are commanded to evangelize, which is different.