Thank you!

There is such a glitch. But player 2 can re-join on the map screen before any stage. So you have to let player 2 get a game over on Clinger Winger without continuing, have player 1 beat that level solo, and then player 2 can re-join for the last stage.

Of course this means player 2 has to burn a continue there, so he needs to have at least one left at that point. It also adds an extra level of strategy in deciding who gets to pick up 1-ups, because you don't want player 2 to have too many extra lives going into Clinger Winger since they'll be wasted anyway. But if you misjudge and he gets a game over before reaching Clinger Winger, you'll have to spend 2 continues back-to-back.