But the next time a creature enters the battlefield, Artisans will trigger again. So at that point, the token that was under my control would be exiled, correct?

Just to clarify, but the token that enters the battlefield will not trigger Faerie Artisans. I’m assuming you mean the next time you, the player with Crafty Cutpurse, plays a creature it will trigger Faerie Artisans and yes your first token you got from Artisans will be removed.