Don't worry about completely saturating the BJT. People make a big deal of it, but it's not necessary, as long as the voltage drop is low enough for the load to work and not result in excessive power dissipation in the BJT, then near saturation is fine.
The 2N4401 is specified hFE > 100, when IC = 150 mAdc and VCE = 1.0 VDC, which would be an IB = 1.5mA, giving a power dissipation of just 150mW. I suggested a slightly higher forced beta of about 50, IB= 3mA, just to be sure, especially at lower temperatures. If you're deriving the base current from the 12V supply, using a linear regulator or resistor, then an IB of 20mA would use 240mW, which is more power wasted, than you'd save with a lower on voltage.