In my mind this is a balk.
NFHS 6.2.4 (b) and (f)
ART. 4 . . . Balk. If there is a runner or runners, any of the following acts by a pitcher while he is touching the pitcher’s plate is a balk:
b. failing to step with the non-pivot foot directly toward a base (occupied or unoccupied) when throwing or feinting there in an attempt to put out, or drive back a runner; or throwing or feinting to any unoccupied base when it is not an attempt to put out or drive back a runner;
The pitcher did not step directly toward a base. His first move was to second then in mid motion to 3rd. By the description was not a direct step.
As far as crossing the back of the rubber...
f. failing to pitch to the batter when the entire non-pivot foot passes behind the perpendicular plane of the back edge of the pitcher’s plate, except when feinting or throwing to second base in an attempt to put out a runner.
Once the non pivot foot crossed, the pitcher was committed to 2 actions. 1. Stepping directly to second or 2. pitching to the batter.
The pitcher would have been correct to step toward second... fake the throw there then throw to 3rd.
Army 54 is correct... the base need not be occupied at the time of the pitch if the runner was advancing to 3rd on the motion.
Now imagine this scenario... a RHP runner on 1st. As RHP lifts left leg, runner breaks to 2nd on 1st move. RHP spins and throws to 2nd ... not a balk provided his left foot did not completely cross the back edge of the rubber. Have yet to see that in an actual game, but have played it over and over in my mind - and can't see where there is a rules infraction.