Is it bad to short a DC motor for a long time?
Shorting a motor by itself is no problem.  It only becomes a problem if external forces turn the motor shaft fast enough to cause too much dissipation in the motor.  When the motor is shorted, all the mechanical power from external things driving its shaft ends up getting dissipated in the motor as heat.  As long as the motor can handle that heat, there is no problem.

It should take no power electrically to keep the motor shorted.  You can keep a FET on across the motor using no power.  It's only when switching that FETs take control power.

Make sure that the wussy little plastic gears in the yellow part can handle the stresses of the external torque working against the shorted motor.  Of course if those torques are within what the motor itself is rated to put out, it should be fine in theory.  The reason I only "in theory" is that on cheap motors with plastic gearboxes optimized for price, the motor's full torque for extended periods of time may damage the device too.