There are several reasons why this doesn't work. CW hits on the attack element, but DTF and Vader operate on different levels.
Die results --> Damage --> Cards. An effect can create any of those things, and you follow the flow through to the end point (damage cards). Once you're at a step, you don't go backwards.
Draw Their Fire moves a die result from one ship to the next. Vader creates damage. So Vader never has anything to do with the die results - there's none rolled, and he doesn't create some pseudo-result. It's just damage, so there's nothing for DTF to latch on to.
As a corollary, this also explains why DTF is useless against Proton Bombs and Seismic Charges. Both effects skip the results and go straight to either damage or dealt cards.