Simulating 1080p pan shot by cropping stationary 4K fisheye footage? I want to achieve a panning shot, but I can't move my camera. The camera is stationary for reasons outside of the scope of this question.

Can I throw on a fisheye lense, film in 4K, and then process it to appear as if I'm panning the camera from 1 corner of the frame in the original footage to the other side of the frame? I don't think this is the same as 4K live cropping, as I imagine I would need to do some processing to invert the barrel effect. 

Does this make sense mathematically? Will the result look any good, or will it be "obviously wrong"? What is the best way to get a good result?
Yes, you can do this successfully. I did it with an overhead 4K action camera with a fixed fisheye lens (YouTube video here).

You'll want to remove the distortion in software to "square it up". Then you have some room to zoom in and pan. However, you effectively lose some resolution with the lens correction. You can only zoom in so far before it's noticeable.  Obviously it's not as crisp as nice 1080 video, but it may be perfectly fine depending on your requirements.