Do you have a link for the rule that says an American citizen cannot pay for WH tours?
Title 18 United States Code section 209 makes it illegal to accept compensation for federal salary or to pay someone over and above their federal salary. Federal wages and salaries are to only be paid by the government.
Since this statute appears in the bribery section of the law, I figure it was put there to ensure someone getting bribed did not say they were paying someone’s salary! Of course intent and what is it used for are two different things, witness the RICO laws where laws originally intended to knock out organized crime have been extended to other groups and matters.
It is as close to what you ask for as I can find. Whether it is actually prosecutable or not, it would at least cause someone some grief in this instance to pay someone IMO.