When thrown, the pitch must be traveling at least as slow as 15ft/s (to give him time to run to the plate and then talk). At the same time, it can’t touch the ground before crossing the plate, or be ruled an automatic ball, so it must actually be thrown upwards at least as fast as to resist the force of gravity (as g =32ft/s^2). Since the pitch crosses the plate at approximately the same height it was thrown, we know when it left Bunny’s hand, it was traveling at about 64ft/s. Neglecting air friction and the effects of spin on the ball:

0s: 64ft/s up
1s: 32ft/s up
2s: 0ft/s up
3s: 32ft/s down
4s: 64ft/s down

64ft/s = 3,840ft/m = 230,400 ft/h = ~44mph

Therefore, he throws the pitch in the air at about 44mph and possibly quite slightly towards home. In the time the toss gives him behind the plate, he begins to chatter. In his three seconds of yelling, he’s able to cause the ball to accelerate extremely fast. We can estimate the speed of the ball given the force applied to Bugs while catching it. If, as seems reasonable, we figure he weighs 80lbs, the force to throw him directly into the backstop and do significant structural damage to that backstop can be estimated ("Estimation of pitch speed through re-creation of secondary observations using weighted mannequin and riot suppresion weapons," Zumsteg, 2004). We are able to figure that the pitch was traveling at least 150mph and possibly much faster.