Is there a way to re-invoke #ajax on forms when the form itself is returned with ajax?
I found the answer!

I simply was asking the wrong question. What I needed to do was re-attach the drupal behaviors with Drupal.attachBehaviors().

I added this code to my JavaScript code.

jQuery('.ajax-processed').once().ajaxSuccess(function() {
    Drupal.attachBehaviors();
});


Voilà! Everything works as it should.