In principle, I don't see any problems. Your heating pad uses 12V and 10W, which means it would be consuming slightly less than one amp of current. The MOSFET you linked can go up to 5A (needing a heat sink with over 1A).
You could use PWM (there are 6 pins on a Uno that can do that) to provide pulse-width modulation to control the amount of output you give to the heater.
You would, of course, need to measure the heat of the device to work out when to adjust the PWM output. You might look up using a PID algorithm to adjust the PWM output depending on the sensed heat input.