By the way, you already have at least one 120v circuit by using that big cable and one of the hot wires. Get rid of the 50 amp breaker and feed one wire with a 20 amp breaker at the main panel. Put a J box in the bath and hook up your 12g romex to that one wire and the ground and neutral.
You also have the second 120v circuit by duplicating that with another 20 amp breaker at the panel and sharing the neutral, but the guys here will probably scream.
Anyone else want to comment on whether or not this is a good way to accomplish my goal?