Temmie would be some sort of "chaos" class.
She only has one move, and it's either going to buff you or debuff your enemies in some bizarre way. Who knows? Sometimes it'll heal/rez a teammate, or deal massive damage to a boss. You can't control the move, and neither can Tem.
The catch? There's a one-in-five chance that she forgets she's even fighting, and will spend that turn pondering a ball of yarn.

What would the other classmate's abilities be?

Noelle seems like offensive buffer (Cuz Susie is the offensive powerhouse).

Berdly a debuffer.

Catti a defensive debuffer.

Jockington makes people tired.

Snowy and Mk are a combo that specializes in TP generation.

Temmie is joke character who joins your party but option to select her is outside the menu, so you can't click it.