This particular version of Spidey starts with a 3 damage, and keeps it for his first 2 clicks. His third click is the best for Pouncing, as it is the only one with both Leap/Climb and 2 damage - he drops to 1 damage with CCE on his fourth click.
So this particular Spidey is probably not a godd choice as a Pouncer. On the other hand, your proposed strategy of assigning Double Time might not work either, as he is a transporter and Transporter don't take push clicks from Double Timing...