Hi,
PWM isnt as good as linear for this application, believe it or not. That's because with a bulb in it's low resistance region you still have to bang the filament with high current pulses with PWM, until the filament starts to heat up. That's not better than turning it on fully. To get PWM to work in this application there would have to be an energy storage element as well, namely an inductor. The inductor would provide a linear current to the bulb while the PWM bang bangs from 0v to 12v.
A linearly controlled MOSFET for this application is not a bad idea. For many applications we would never want to control a MOSFET or other transistor in the linear region, but for this app we have an operating time that is much much shorter than most applications. It only has to control the MOSFET for a second or so not for say 3 hours. So the transistor is in it's linear region for 1 second and fully off or fully saturated for all other times. Since the average power dissipation highly depends on the operating duty cycle, the MOSFET would have plenty of time to cool down. Perhaps a small heat sink. If the lights have to blink on and off then the heat sink would have to be larger, but for normal operation it should be just fine.
For example, say the lamp draws 10 amps at full power with 12 volts. That puts the max power of the transistor at around 30 watts (max power point). But lets assume even higher, lets assume it is 60 watts. Normally a heat sink for a 60 watt device would have to be large or cooled with forced air. But for 1 second of operation at 60 watts (and that power is not present the whole time anyway) even a smaller heat sink would not heat up that much.