Um... TF's Pick A Card is on-hit isn't it? You know what that means...
"Modifies next attack".
Lyte posted a very simple rule-of-thumb: Would you get more procs if you attacked faster? Then, yes, it works. Which means that most likely TF's passive would work.
Would you not get any more procs? Then, no, it does not work. Pick A Card would not give more procs, so it does not work.
This goes for pretty much most attacks you could think of.
Ok, well... it's not as fun but he can still combo it with Stacked Deck and Lich Bane. You could have some nasty AoE burst if you times it properly with his other abilities.
Roku has hinted that there will be methods to reduce Flash's cooldown. Perhaps the introduction of a Blink Dagger-esque item?