Best answer yet and most thorough. Thank you
Let me throw a wrinkle in... Foreign Emoluments Clause...would that apply since he’s in the Royal Family?
Assuming his mother doesn't take care of it first, he'd still be a member of the family but not eligible to claim any sort of nobility title or privilege that comes with that title with the US government. This was probably a bigger deal two hundred years ago when nobility was more than a historical curiosity and subject for gossip columns and gaudy tourist attractions. Used to be that offending foreign nobility was a big deal. They were big-time celebrities and mistreating one could be cause for wars. We have more cable channels now and more celebrities to ogle, and war isn't nearly as appealing unless there's a dollar to be made. For all intents and purposes, there is no real thing such as "international law" in that there is no superior legal system that can compel sovereign nations to obey each other's laws, so the UK (assuming it still has a royal family in thirty some years) could say he's in line to the throne and considered nobility, but so long as he didn't claim diplomatic immunity as a member of the royal family or something that would show him acting on it, it's just sort of a curiosity.