seems to suggest that the limit is 1A.
Yes that is the current limit but the current limit is not the limiting factor. That is because you could have 2V dropped across the regulator or 4V and that doubles the power. 4W is a lot of heat in a small package.
As to peak power that is not as important as peak current. With peak power you can smooth it out to the average, so you can often stand brief pulses of power, many times over the continuous rating.
Applying a heatsink is simple, just clamp or glue a piece of metal on the regulator. The metal should have as much surface area as possible to have a low thermal resistance to the ambient.