I have dived with some Nat Geo Videographers and loved their comments "Set the Camera to auto and shoot with the right unit". I was white balancing every few meters / change in light. These guys were just shooting.[/
Still camera's, from what I read and hear, tend to need a lot of tweaking to make good video.
I currently have the Sony A1p, which in 2006, was the entry level professional video camera of the day. It still does wicked footage in the right conditions. Brilliant Macro night or day. Brilliant wide angle in good light. Dies in low light. The Sony in the cave performs beautifully with clear and crisp up to about 5 meters distance. The GoPro is blown out and colours are all wrong. Go out to 10 - 20 Metres distance and the Go Pro Footage is by far the winner.
I mention the GoPro as its is supposed to be a toy supplementing my main unit but in caves, it made my sony obsolete.
For lights I'm running L&M Sola 4000's.
Money - $7K - $15K for the right unit. The Sony FS100 would be in the budget but the Sony FS 700 looks awesome. But do I want to spend that much????
I'm guessing these Nat Geo shooters didn't tell you that they don't have to do the grading/color correction and just hand in the clips. Could you tell us which Nat Geo shooter says shoot auto? I'm really interested to know.
Anyhow, if you are looking to upgrade to full HD with better noise, on that list you have, I'd drop the Nikon D600/800 for noise as they are just no match for the 5D3/6D in terms of video noise, especially in the high ISO range. The 6D is especially good with video noise. Of course, the 5D3 shoots RAW now via a firmware hack. The 6D can't really match it because it uses SD cards. The 5D3 can now shoot 8 bit 422 uncompressed out of the HDMI, and the difference is noticeable for water column banding, but not significantly so.
Obviously noise is only one aspect of video quality, there's color, resolution etc. The FS100/700 are nifty cameras hampered by WB problems in certain conditions, but the video coming out of it is very clean and good resolution. However, only the FS700 will shoot RAW with a RAW recorder and that will bust your budget by quite a bit, once you include housing etc.
The BMCC (v1.0) is a nice camera for good light conditions but it gets pretty noisy past 800ISO. I wouldn't bother with it since you'll have to deal with RAW .DNG files as well. The new BMCC promises to be even better resolution but the noise issue will still be there.
The GH3 is a nice little camera and for video with good lighting, it probably is the better camera. It does get very noisy and the 8bit banding is not as good as the 5D3 for water column shots. The noise and resolution gets pretty ugly after 1600ISO, especially in the shadows, typical of the diving you will be doing. It'll still be better than your A1 by quite a bit.
As you can see in Berk's video @ Cenotes, powerful lights and a good low light camera like the 5D3 can produce great results. I personally think if you only want to shoot video, can get use to a DSLR (shallow DOF, no autofocus) way of shooting, the 6D is probably the best low light performing camera there is for under $10k. It's not the best looking but has the least noise. If you liked the HDV resolution of the A1P, you'll love the 6D ALL-I codec and the low noise at even 6400ISO.
I'm pretty sure with a housing, camera and a few good lenses, you should come in under $11k. Like others have suggested, look into more powerful lights, perhaps even going for a more narrow beam so there's more penetration to beat the shadows.
Choosing between the GH3 and 6D, I'd still pick the 6D myself, just because of the noise performance. It's really good even @ 12800ISO, which is unheard of just 2 years ago. So choose what is important for your needs and go with that.