Yeah, actually it's all about the friggin' United States Constitution...if you want to change it, that takes a constitutional amendment...
You're incorrect. It doesn't take a constitutional amendment. An act of Congress could change it.
The constitution states: "“No Title of Nobility shall be granted by the United States: And no Person holding any Office of Profit or Trust under them, shall, without the Consent of the Congress, accept of any present, Emolument, Office, or Title, of any kind whatever, from any King, Prince, or foreign State.”
"....without the Consent of the Congress..." means that congress can approve a president taking any of these things. They could write terms in advance of what could be acceptable. Just like they write the variable terms regarding Patents and Intellectual Property law that's required per the constitution.
As an aside, when Obama was awarded the Nobel prize for peace which was worth over a million dollars, Congress approved his acceptance of it, even though he gave the money to charity. In any case, I think the "bribe" worked anyway as he acted as a very weak president, trying to show he "deserved" a peace prize. So in that sense, the emolument worked even though he passed on the award money.