Exiling tokens with conjurer's closet
The token will disappear, and it will never enter the battlefield.

You are correct that state based actions are only checked after the entire ability resolves. However, there is an additional rule that applies here: rule 110.5g says


  A token that has left the battlefield can’t move to another zone or come back onto the battlefield. If such a token would change zones, it remains in its current zone instead. It ceases to exist the next time state-based actions are checked; see rule 704.


So once the token is exiled, it stays in exile even when the Conjurer's Closet instructs you to return it to the battlefield.

So, the use of the phrase "that card" on Conjurer's Closet is not the reason that it only returns to the battlefield if it's a card, but it's a hint that that's how it works anyway.