I find that villain needs to fold a king more than 2b/(1+b) of the time to be + or neutral ev if we bet b when we held a queen.

It would appear that villain needs to fold a king a fairly large amount of the time for a bet of any size to be +ev though and this seems pretty hard to accomplish in such a game if we always bet every queen and ace.

Now if we only bet a queen x% of the time...hrm...