You will not burn out your motor. by using a larger mah rated battery, as long as the voltage is the same.
I use 160 mah Hyperion batteries in my MSR, I think they are 20c or 25c.
My MSR pulls 1.6 - 1.7 amps at hover, the 12c 120 mah batteries should only put out 1.44 amps, so they are under a little stress, but they don't seem to get warm.
I would not want to carry a heat sink, the MSR doesn't have a lot of excess lift.
The motors are a simple brushed motor, 100 flights is probably all they can handle before the brushes are gone.
I need to fly mine more, but every time it lands, my dog tries to eat it.