Could Trump insist a request to be 'Knighted' on Inauguration Day!?

That's a British custom. They would NEVER do that to Trump. But perhaps we could make him the official Dunce. I think he'd look about right in a dunce's cap.

He would not be addressed as "Sir".

U.S. law prohibits U.S. citizens from accepting knighthoods.  When U.S. citizens are "knighted", they are given only "honorary" knighthoods.  "Honorary" knights are not addressed with the word "Sir" followed by their name.  That style is used only for the Queen's "subjects" (citizens of places where she is the Queen, such as the UK, Canada, Australia, New Zealand, etc.) and not for Americans.

America doesn't grant titles of nobility.

The queen would laugh at him.

Nope. Foreigners cannot use the title even though they receive the honour. Stormin' Norman Schwarzkopf is a Knight of the Realm. Rightly so, but the title is silent.

its up to trump

He cannot be knighted.

He could request it, but of course that would only make him look even crazier than he already does.