EXCEPT his name is EDWARD William Proxmire so the clue would be “Former Senator Edward W.” according to the rules of the game.
So you are wrong sir, wrong and FTR this is now the second time you’ve changed the name in the clue to something close to answer my post with an incorrect answer (the other time you changed the middle initial from A. to R. to answer it.)
Still in play
Former ex-Senator and holds a unique distinction: William
The “ex” Senator is a key part of the clue. In other word he is more famous becaus he is an ex-Senator than for being Senator.
The “unique” distinction is one that cannot be matched and pertains with why he is famous.
There is no middle initial since he did not have a middle name.
His legal first name is not Edward.
So he’s a “former ex-Senator”, does that mean he was once in the Senate, got voted out of office (or otherwise left) and then was re-elected? Or is he “former” because he’s now dead?
ex means did not leave office by traditional means i.e. died or voted out of office.
Seems like we’ve stalled.
Secretary of the Treasury Salmon P.
Chase
Former White House Chief of Staff Rahm I.
Emanuel
Governor of New Jersey George B.
McClellan
Senator James
Hint: He was a Senator for three different states.
Senator William Blount.
The only Senator to be impeached. The Senate rejected the impeachment as a matter of legal principle (arguing the House of Representatives didn’t have this authority over the membership of the Senate) but kicked Blount out on its own.
Senator James Buchanan, Jr., who would later become president, had no middle name. I don’t think he matches your hint, but if you had meant that to be a requirement perhaps it should have been labeled as such.
President John C.
Coolidge (Full name = John Calvin Coolidge Jr.)
U.S. Secretary of Housing and Urban Development George W.
Romney
Governor of Ohio Richard F.
Celeste
Ambassador accredited to Monaco Craig R.
Double Points for Little Nemo 
Stapleton
U.S. Supreme Court Justice Ruth B.
Ginsburg
Secretary of Labor Arthur
Goldberg
Postmaster General Francis
Granger
Confederate Secretary of State Judah P.
Benjamin
Emperor of the United States Joshua A.
Norton
Ambassador John C.