I see no particular constitutional problem with a U.S. state switching to a parliamentary system of government. Politically, though, I think it’s very unlikely. Barring a huge scandal or a crisis, people tend to be very reluctant to tinker with the very bedrock structure of American government at any level.
France and Russia also have non-monarchial parliamentary systems.
Japan has a monarch, Emperor Akihito. The postwar Constitution specifically does not designate him as head of state, but rather as a national symbol, although he kinda sorta does act as head of state in an even more limited way than in the UK. His assent is not required for legislation to take effect, for instance, and the nation’s courts don’t mete out justice in his name.
I would characterise them as hybrids. The Presidents in both those countries have a lot of power, much more so than Her Majesty, or the Presidents of other parliamentary republics, like the Presidents of Germany and Italy. (As touched on briefly in my earlier thread about the May 1946 referendum in France - the President of the 4th Republic was much closer to the figurehead role than is the President of the 5th Republic.)
That’s assuming that the National Assembly tried to proclaim independence by statute. If they did it by resolution, the Lt Gov would have no power to reserve it.