If a character with two action tokens and willpower has its willpower outwitted, does that character take pushing damage?

I'm sure there'll be a more official answer, but if it wasn't outwitted before the second token was applied, then no pushing damage should occur. That's the way I've always read it and acted accordingly.

No.
Pushing damage is only checked at the end of an action in which a second token is applied: If the character has some way of ignoring pushing damage then (Willpower, Power Cosmic or Quintessence TA), then pushing damage is ignored. Afterwards, it is too late to ignore that damage.

As a comparison, it would be kind of like saying "I'm going to Outwit your damage reducers for the attack I made five minutes ago"...

You cannot outwit in the middle of an action. You have to do it before an action.

However, there are some unique instances when this would happen.

For example, you have 1 action token. So you "push" your character (with willpower) to make an attack. If it is a critical miss, then you take 1 damage, which makes you lose your willpower. Now that your action is done, you apply push damage. Does he have willpower? Nope, therefore he would push damage now.