How fast will a resistor heat up under high power?
Far easier: use an automotive 12 volt, 2 ampere incandescent lamp ("bulb") as a series resistor. This has a number of advantages:


For short duty cycles, the cold resistance is less than a tenth of the hot resistance of the lamp, 6 ohms. This provides more current to the circuit.
If the circuit draws too much current, resistance rises, limiting the total to less than 2 amps.
No heat sink is needed, if the lamp is in open air.
You have a visual indication of current drawn.
These lamps are cheap and readily available.


If 2 amps is a bit too much, add a small series resistor, which does not have to handle high power, since most is being dissipated by the lamp.



You'd need to know the mass of the resistor, the specific heat capacity of its materials (e.g., ceramic) and the duty-cycle of the current (percentage of the time it's on). That said, the resistor would work for some seconds or minutes, and then act as a "slow-blow" fuse.