I think it's a leftover from the days of Roman occupation. Fiat di seems to be used as a brusque but friendly brush-off parting, simply expressing "OK, I'm done with you. Good-bye" I believe that after watching how the phrase is commonly used in Bavaria. Sometimes it is said with kind and loving attitude, of course, but it is also often used as a clear closure to an interaction, as from a waitress or store keeper, as in, "our transaction is done." There is nothing about Fiat or the Bavaricized version, Pfiad, that sounds anything like most of the words that people have speculated are roots of this. If it was from Fuhrt or Behut or whatnot, there would be likely some neighborhood somewhere in southern Europe where you could still recognize something more similar to the actual root. I think the actual root is the actual current word, fiat, in Italian. It means "Let it be done" as described in this reference in Wikipedia (sorry, I know some of you don't like Wikipedia, but just because a speaker makes some mistakes, does not mean that the speaker has nothing valuable to teach you.) The term fiat derives from the Latin fiat ("let it be done") used in the sense of an order, decree or resolution.  Fiat is the third-person singular present active subjunctive of fiō ("I become", "I am made").  Schueffel, Patrick (2017). The Concise Fintech Compendium. Fribourg: School of Management Fribourg/Switzerland. Archived from the original on October 24, 2017.
So fiat di is simply "let it be done/resolved with you", just as fiat eich or fiat euch is, likewise. So fiat is from the Italian/Roman, and di and euch, from the German/Bavarian.