So as an owner of these two cameras, here are my thoughts:
First, the bmpcc4k right now is shooting blackmagic raw, and unfortuntely blackmagic raw is not true raw, it's partially debayered in camera, and for some reason on the p4k and p6k camera's braw is especially soft and mushy, i would say that the hd cinema dng from the bmpcc original is comparable to the p4k in braw. You really isn't getting the full 4k resolution out of the blackmagic raw files, especially due to their high frequency detail filters that eliminate fine details such as hair, fabric textures, small texts, etc. This was done as a way to denoise the footage in camera, and was sharpened as well, creating a mushy image, with a green fringing commonly present in the prores and blackmagic files on the pocket 4k post firmware 6.2, I originally thought that it was my lens's fault, but turned out it was braw, as the cdng stills did not have those green fringing.
Second, the bmpcc4k's colorscience is intresting, it's a lot easier to manually grade by hand in terms of getting a pleasing image, due to the fact that blackmagic seemed to have aimed for skintones instead of accurate primary colors. So even though the skintone is very healthy, the greens are very brown. And if you attempted to fix those greens in post, the afformentioned mushiness and green fringing becomes extremely obvious. I owned a lot of pocket 4k luts, like the p4k2alexa, p4kalex by emotive color, the buttery luts, and filmconvert nitrate. Which all aim to match the pocket to arri alexa or a cineon log film scan of different film stocks. But since the alexa has very vibrant greens, even though the color look great on the p4k2alexa and p4kalex, the mushiness and green fringing is exaggerated.
Third, I would say that the lowlight performance of the p4k, is about 0.66 stop noisier then the s5, and the exposure lattuide to be a stop lower then the s5 when having highlight recovery turned on (the s5 does not have highlight recovery in blackmagic raw), due to the fact that s5 is better then any camera under 10000 dollars when it comes to usable dynamic range and latitude . Speaking of which I do own a blackmagic video assist 12g to record both prores and blackmagic raw on the lumix s5. The lumix s5's blackmagic raw is somehow quite a bit more detailed then the braw out of the p6k, and the 4k apsc crop blackmagic raw on the s5 definitely has more detail then the braw out of the p4k. But in terms of natural texture it still does not hold a candle to the normal prores 10bit footage recorded on the video assist. But the weird thing about the lumix s series of cameras is that their noise reduction seems to have some waffle shaped artifacts, which is apparent when you push the colorgrade in post at iso4000 and above, with a bit of banding present on bright areas that should not be there on normal prores 10bit as well. I have reported these issues to the panasonic team so hopefully they will get fixed soon, in which case I would whole heartedly suggest shooting prores, as it has much better performance and have smaller file size, but until then i will be sticking with the blackmagic raw on the s5.
Fourth, almost all modern mirrorless and dslr have a very nice and strong IR cut filter built into their sensor, which means that even with high levels of ND filtration the skin tone, and blacks in the image stills normal. However, the blackmagic cameras have always had bad IR cut filters, it is better on the p4k then on the original pocket, but outside even without any ND black fabric tends to turn purplish, and skintone becomes quite pink as well. This is not completely fixable in post. Thus a external IR cut filter would be needed on the p4k. I first purchased hoya ir cut filter, which does filter ir light well, however the filter's coating is very soft. So after a bit of usage the filter was very scratched, and if you shoot any wider then 35mm in fullframe equivalent, then you will see green vigenetting on the edges. Which is a pain to correct in post, the only IR cut filter that I recommend is the heliopan IRUV cut, which doesn't have the green vignetting issue and has a very strong coating just like a normal lens filter.
In terms of usability I prefer the p4k, the dials position just makes sense and since you don't need a video assist on top of the camera the balance is great. The motion on the p4k is also very smooth, smoother then the s5 in the same framerate and shutter speed.
I would also recommend you to get s1alex for the lumix s5, because it converts The V-log from the s5 to arri log c, which means that it's much easier to get a very pleasing colorgrade from the s5. The s5's internal 150mbps codec is the best internal codec in mirrorless cameras, since it preserves color details in the shadows much better then the XF avc codec on canon cinema cameras and XAVCS-I codec in sony cameras. But it's still only 150mbps, which means that it does have some blockiness if you have very fast shutterspeed and a lot of moving stuff in the frame. But that's why you get a external recorder for prores.
In the end I recommend the s5 a lot more, not only does it not have the mushiness and fringing issues of the pcoket 4k, it also is a fullframe sensor, thus having amazing dynamic range and better lowlight. It also can cover the entire image circle of your FD lenses, thus I recommend it more.