Can I new-game-plus Borderlands: The Pre-Sequel with a different character?

Nop. You have to start from 0 with another character to be able to play in: "new-game-plus mode"

I believe there are save-game editors that would allow you to take a newly-created character and change their level, inventory, and story progression so that they'd be on the second playthrough.

However, I wouldn't recommend it - you'd be playing against harder enemies with a character that you were less used to.

If you just want to pass along stuff, I seem to recall there being a stash in town that'll let you pass a small number of items between characters, so that will give your new character a bit of a leg up.