Can a Tenacious Z be Played After Drawing Two Cards?
No.

From the Pretty Pretty Smash Up rules, page 10:


  On your turn: This means during the normal Play Cards phase of your turn.


The turn order is:


Start Turn
Play Cards (normal time for plays)
Score Bases
Draw 2 Cards (and then discard)
End Turn


By the time you're drawing and discarding, you've missed the window of opportunity for Tenacious Z's special. If it wanted to override this limitation, it would have to be explicit (e.g. "during any phase on your turn").