Why not just custom white balance whilst descending to target depth? E.g. white balance at say 20 meters and then continue to 40 meters. This way you eliminate some of the blue or green color cast, but are not yet boosting red to extremely noisy levels.
You can certainly experiment with this.
With cameras that have a limited range for white balance adjustment (e.g. Sony cameras), this is more or less what you are forced to do since trying to set white balance too deep will likely result in an error.
Still, for my tastes, the results of white balancing at a shallower depth and then filming deeper are a bit too greenish-blue. It's not just the color itself, either. A large part of the contrast we perceive in an image comes from color contrast. For example, reds are much more noticeable when placed against green, blue against yellow, etc. So, images that include the warmer tones tend to have more contrast and are more "poppy" than ones that are all just shades of blue and green.
My preference is to use artificial light (strobes or video lights) when filming below 10 meters, and keeping the white balance set to the temperature of the lights. I find that it producing much more pleasing results than trying to shoot with only natural light.
Does it accurately reflect what I saw on the dive? No. But for me, photography isn't about striving for the most accurate version of reality. It's about producing beautiful images. Accurately capturing reality is not as straightforward as you might think, anyway, since cameras see the world very differently than our visual system processes the world. Physiologically, most of what you "see" is actually a very detailed model constructed by your brain based on your past experience, based only partially on the limited, low bandwidth real-time input from your eyes.