Is there a 4K action camera solution that does NOT have perspective distortion?
There are two major advantages to recording fisheye distortion rather than rectilinear distortion.  First, it is optically easier to do, which means one can make lenses that are better in other metrics (such as chromatic aberration, resolution, vignetting, etc) while remaining small enough and light enough to fit into the body of an action camera.  Second, the relative size of an object doesn't change radically as its moves from center to edge when the camera is not under tight control.  For that reason, it is possible to define a crop area that follows the subject as it bounces around the full frame and then use motion tracking to create a relatively stable view of the object, which can then be de-fished.  If it were rectilinear, the stabilization process might have to do some major zoom and tilt corrections as well, which would be very distracting.