Running DC motor at higher than spec voltage, but less current than max
Your aversion to MOSFETs that work on 6 V or less gate drive is silly.  The simplest answer is therefore to use a 6 V supply with sufficient current rating to fully run the motor.  You are then running the motor in spec, so there should be no problem.

For completeness, if you only had a 12 V supply for some reason, there are some options:

Just do it.  Since you are only running the motor for 15 seconds at a time with long rests to cool off, it will probably not get hot enough to suffer damage.  This will be especially true if the motor can rotate nicely.  That will cause it to develop back EMF, which effectively subtracts from its supply voltage.  The motor will actually draw less current at 12 V when turning fast enough than at 6 V when stalled.  It's the current that matters to heat dissipation.

PWM with 50% duty cycle.  Use a fast enough PWM frequency so that the current changes little during one pulse.  Since you can use "dumb" 50% fixed PWM, there should be no problem running it at 50-100 kHz.  That's still plenty slow enough to avoid most switching losses.  Make sure to put a Schottky diode in reverse across the motor.  That will keep the switch from getting fried, and give the current a path to keep circulating during the off phase of each PWM pulse.



Your 12 V supply is inapprorpriate because it can't provide enough current.  When the motor starts, it will try to draw a lot of current.  The supply voltage will collapse, or the supply will temporarily shut down.  Either way, the motor won't get going.