Dropped third strike + foul tip

Question
I encountered this situation while umpiring and still am not completely sure of the proper call.  1-2 count on the batter, bases empty, two outs.  The pitch bounces a few feet in front of the mound.  Batter swings and foul tips it.  Catcher catches the foul tip cleanly.  The catcher rolls the ball back to the mound and heads to the dugout, while the batter runs to first base.

The two rules I was trying to reconcile:  (1) a third strike that bounces on its way to the plate is a dropped third strike (even if caught cleanly by the catcher after the bounce)and (2) a foul tip caught by the catcher on third strike results in strikeout.

Does the caught foul tip change the fact that the pitch had hit the dirt before the swing?  After the swing, it went directly into catcher's mitt.  Any thoughts?

Thanks!

Answer
Great question...talk about an odd situation. If I were making the call, I'd say that under that exact scenario, the batter is out since the foul tip went directly from the bat to be caught by the catcher. If the batter would have missed the ball entirely on his swing, then the batter should be able to take off to 1st base on the basis of a dropped third strike. I'm pretty sure that's the correct interpretation, but good luck finding that scenario in the rule book!