"Shooting the still with a fisheye lens and the video with a rectilinear lens would present alignment issues, I fear."
I've struggled to get seamlessly-embedded video clips lined up in KRPano. I achieved this more easily in 3DVista VTP, but I think my experience may still be relevant in terms of your concerns...
I found that the video clip MUST be shot with a rectilinear lens in order to fit into the scene properly. Anything shot with a fisheye lens was distorted in the final result and precise alignment was effectively impossible to achieve. I had moderate success with defishing tools in video editors but I never found anything that has the precision of a RAW photo editor. In the end I concluded that the only serious way to capture video clips for integration into a 360 photographic scene was to shoot with a rectilinear lens, regardless of lens was used to capture the 360 photo content. Just keep it on the tripod and pano head.
(Keep in mind that whatever you shoot with will be entirely reprojected as part of the equirectangular stitching process! You can shoot a pano with a rectilinear lens or a fisheye lens. The reason we typically use fisheyes is only because we get a wider angle of view for a given focal length. My 16mm fisheye captures more than my 14-24mm rectilinear lens even at 14mm.)