In this case whatever the rule is the joker would always be equivalent to a non royal king (the opponent has only one movable piece, a king).King+joker versus bare king. The ending of king+non-royal king versus king is an easy win, but if the joker is unable to deliver check (which is one possible interpretation of the rules you posted)
I think you are alluding to a subtlety involving stalemate here. If you do not consider a virtual check (a check by a joker before the opponent moves) as a genuine check then there would be only statemate positions.
So I guess for the purposes of checkmate vs stalemate we should consider a virtual check as a genuine check.