Can I get away with a large enough copper plane?
Not likely. That's too much heat concentrated in too small an area.
Also the area required to dissipate than much heat would required so much area it would not be economically feasible.
Here is an example of two LEDs dissipating only 1.5 W each.
The question is: is a heatsink necessary for a 60 second (max)
operating time? Assume that the down-time between operations is larger than the operating time.
So 60 s on and 60 s off. It will only take longer before it burns the FR-4.
What will happen is the PCB temp will increase in steps. When the 60 second off cycle ends the PCB will be warmer than when the previous on cycle began.
Each cycle the temperature is warmer than the previous. That is until the temperature differential between the copper surface and ambient air is so great that natural convection becomes effective enough to where the temperatures stops increasing.
As shown in the convection formula the heat transfer flux (q) increases as the temperature differential increases. By that point the FR-4 is burned.
I do not know your reasoning behind choosing a TO-220 package, but it is not a great choice. You would be better off using a through hole ceramic wire-wound power resistor where the body of the resistor is offset from the PCB.