No violation?
 

Player A1 leaves feet to attempt shot. Defensive player B1 is in guarding position for a block attempt. Player A1 intentionally drops ball, allows it to contact the floor, regains possession and dribbles out to attempt shot.

Officials called this a travel violation. Because the ball was allowed to contact the floor and it was not a shot, wouldn't this be legal - same as batting a ball over your defender and continuing the dribble after the ball contacts the floor?

It's a travel because the player essentially began a dribble after his/her pivot foot was off the floor.

If the player had already dribbled it could also be an illegal dribble.

Avoid extreme temperatures and store in a cool dry place.
 

When an airborne player keeps control of an attempted shot that is blocked and is unable to release the ball and returns to the floor with it, that player has not traveled; it is a held ball. If, in this situation, the shooter releases the ball, then this is simply a blocked shot and play continues. When an airborne player tries for goal, sees that the try will be blocked, purposely drops the ball, and picks up the ball after it hits the floor, that player has traveled by starting a dribble with the pivot foot off the floor, whether, or not, the defensive player touches the ball in the block attempt.

4.44.3 SITUATION B: A1 receives the ball with both feet off the floor and he/she
lands simultaneously on both feet without establishing a pivot foot. A1 then
jumps off both feet in an attempt to try for goal, but realizing the shot may be
blocked, A1 drops the ball to the floor and dribbles. RULING: A1 has traveled as
one foot must be considered to be the pivot and must be on the floor when the
ball is released to start a dribble. The fact that no pivot foot had been established
does not alter this ruling.