All the better for Mewtwo to use Me First before that move. ^^
I'll do my best to prove it now...
Okay, so Bowser was invincible from Mario's attacks cause of the Star Rod. Mario attacks mostly using his jumps and hammer in Paper Mario right? All physical attacks.
Now being a psychic means that one is exeptional gifted in metal abilities. So most likely, a psychic would attack with it's mind. The mind as we know, works through the brain which is a vital organ inside one's body.
Psychic, the move has as a description: "The foe is hit by a strong
telekinetic force. It may also reduce the foe's Sp. Def stat. " The Japanese name for this attack is サイコキネシス Psychokinesis.
So yeah there you have it... Most likely, Bowser's invincibility doesn't work against mental attacks.
As for the current match up:
This has to be the most unfair match so far.
Falco wins, cause the Ice Climbers can't do anything. They may hide, but Falco has bombs.