Roosevelt relied heavily on his advisor "Wild Bill" Donovan in much the same way as Trump does on Rudy Giuliani. Both were personal envoys. And Vernon Waters served as a roving diplomat for Reagan. None of these are official Article 2 officers, but are they considered then White House staff and paid from that budget, or other taxpayer provided budget, for the work they are doing on behalf of the president that would be considered public?
I wouldn't expect Giuliani to be paid for the work he is doing as the President's lawyer to be considered part of that. So just the public work please.