I'm not a technical expert by any means but the way I understand it, is that any white balancing at depth, whether manual or automatic, will introduce a certain amount of noise, because effctively you are 'artificially' inreasing the red channel - and that will add more noise to that channel, because by white balancing you are not adding any more information to that channel. Things should be better in 10bit because of the addl. color info compared to 8bit, so I am having good hopes in the GH5 in that respect
Whether there is more noise introduced on a Panasonic compared to lets say Sony I dont know. But I can tell you from my time with the Sony AX100 camcorder it was introducing plenty of noise if you where below 15 meters on ambient light only. I guess thats just a critical depth fo most cameras where they start to struggle.
Right -- what you're essentially doing when you're white balancing at depth is telling the camera to apply gain to the brightness levels recorded by the "red" photoreceptors. So it's equivalent to boosting the ISO, and will make any existing noise in the red channel more apparent. With the GH4, the noise was quite visible/distracting when trying to white balance around 20 meters (especially if your base ISO was already 400 or 800 to account for the lower light).
This step happens before any of the data is encoded into an 8 bit or 10 bit video file-- white balance is processing applied to the 12 bit data coming right off the sensor. So the final recording format isn't going to make a difference to the amount of noise. The GH5 should still be better overall then the GH4 simply on account of newer sensor technology, using a larger area of the sensor for video and better noise reduction, but the differences are probably similar to the overall improvement in low light performance. DXOMark rates the GH4 and GH5 as essentially identical in terms of noise performance, so I wouldn't expect the GH5 to be significantly better than the GH4 in terms of noise when white balancing. A full frame camera like the 1DX Mark II should be 1-2 stops better, which is what the Backscatter guys seem to report as well.
As to whether a 10-bit 4:2:2 h.264 file will make it easier to correct a bad white balance in post than an 8-bit 4:2:0 file h.264, it might make a bit of a difference, but it won't be night and day like you would get if you were recording raw. During the encoding process that creates the h.264 file, the 12 bit data in the red channel is going to first get confounded with the signal from adjacent blue and green pixels as part of debayering. Then it's going to get compressed from a 12-bit space where 4096 gradations of brightness can be represented to a 10-bit space that only supports 1024 gradation. That alone throws away 75% of your data. Then you end up throwing away the color data for half the pixels when you go from 4:4:4 to 4:2:2 color. So you're already down to 12.5% of the original data (not accounting for errors introduced by debayering based on the wrong white balance) before you even get to the compression stage. Then you add the fact that h.264 compression tends to preferentially eliminate data in the shadows. Your red channel underwater is mostly going to contain data in the shadows if it's not pre-boosted by a "proper" white balance before this stage is reached.
So yeah. If you want to retain information in the red channel when shooting in a compressed video format, you need to make sure that the data is properly baked in by a good in-camera white balance. Since the red channel signal is going to be very weak coming right off the sensor, if you don't boost it significantly before recording it (and more importantly, compressing it substantially using h.264 compression), you'll end up trying to reconstruct what should be dozens of shades of red of various brightness from very little data indeed.
All of which is to say that, when recording to any sort of compressed RGB or YUV codec underwater using only ambient light, the camera's ability to do a proper white balance in the 40000-50000 kalvin range with significant magenta push is far more important to the final result than the bit depth or color depth of the recording codec. Good white balance flexibility will get you nice colors even in a 8-bit, 4:2:0, low bitrate codec. Case-in-point: canon dslrs. Whereas if your camera won't let you set a white balance higher than 9900 kalvins (e.g. all Sony mirrorless cameras that, and maybe some of their cinema cameras too), the codec won't matter. At least, not unless you can record the raw sensor data before debayering and compression.