Briefly, I am recording in 4K (Sony X3000) and doing some lens corrections, specifically using the Sony Deform (to introduce about 5% pincushion to counter the camera's barrel) and then using the track motion feature to zoom in just enough to crop off the four dark bits.
Then I render to 1080P. So only 1/4 of the pixels are being used in the output. So there should be no visible loss of resolution.
The problem is that there is a huge loss of resolution. It doesn't matter what output codec I use.
I have also tried the NewblueFX Lens Correction but it is just as bad, and actually doesn't do a correct correction (it might work for some specific cameras) and introduces corner artefacts. I have contacted them but they don't really understand...
I get a much better result shooting in 1080P and forgetting about the geometric stuff, and accepting a bit of barrel, which on the X3000 is not bad; much less than any other action cam. Especially as the 1080P is 60fps whereas 4K is limited to 30fps, and the smoothness is subjectively very nice.