I don't think it entered the thoughts of those who wrote the US Constitution and its amendments, that someone in the pocket of a foreign power would ever be elected President. Impeachment would leave Vance in office, wouldn't it? Successive impeachments would be required until a sane VP is chosen.
Check your history. Nixon resigned to avoid impeachment but VP had already left after being caught in his own crimes. So Gerald Ford became president.
It's not my history, but I get your point; my scenario assumes 1. that the VP is legally blameless at the point of taking office and 2. that once he takes over as president (no females envisaged at present) his own choice of VP will also still be legally blameless as the next president is impeached.
Please push for King Charles to cancel his invitation. Trump should be no where near royalty. He is a clear and present danger.
He'd need something more than the broadcast bullying of a head of state to cancel; British Royalty entertains heads of state with appalling human rights records, without batting an eye.