Might be better to use a panohead to take a 360 cylindrical pano with lots of overlap, create many spread out control points, optimise, and then save the lens parameters. This will give an accurate horizontal field of view, (equivalent) focal length, and the distortion values.
When you say, switching to rectilinear projection, you mean the output projection, not the lens mapping (should be a rectilinear lens anyway)? Looks like it's a problem with the field of view of the output causing the rectilinear projection to stretch the periphery. Some of your images are badly places causing this, it would seem.