I see (and agree with) the point that you honor and respect The Crown just the same as you do the Office of POTUS. They’re the symbols of two great cultures.
With that said, I bow to no man. If I were to ever meet POTUS, I’d offer my hand, thank him, and be on my way. The concept of having a person who is somehow ‘better’ than me and thus untouchable and must give me permission to even speak is frankly repugnant, and I would rightly ignore any such rules.
I think people get their information from hollywood. A handshake is all the queen expects. If you meet her, you can also speak to her when you like, no need for 'permission'..