I like this idea a lot. Gleason as Nixon's populist attack dog instead of Agnew? As a representative of square, Middle American pop culture, he'd be perfect for the role. He'd have some better lines than "nattering nabobs of negativity," at least.
Shamefully, I didn't know about Nixon and Jackie Gleason's friendship until I read The Secret History of Twin Peaks a few months ago (which is a trove of things that sound like AH but aren't - to the point that you start to wonder how little original thought Mark Frost put into the book in the first place).