Inrush current from DC motors causing Arduino to reset
I'd be inclined, given the huge headroom (15 - 9 = 6 volts), to just use a 2nd regulator for the MCU.

Make sure the motor-regulator (dissipating 1.5 amps * 6 = 9  watts) is well heatsinked.
Or use a switching_reg for the motors.