There must be a secret language of power plays and protocol that is common in American politics. In other words, Donald could very easily scribble a check out and fund his own inauguration w/o batting an eye. So why doesn’t he just do that?
Maybe he, like many very wealthy, become very creative in learning how to do things without depleting your own personal funds. I doubt very much if Bill Clinton or Obama paid very much toward their inaugurations. Maybe this is supposed to be an early opportunity for Lobby Groups to make themselves known to the victor in the White House.
Washington was probably the last president to pay out of his own pocket. Even the Roosevelts and JFK took donations.