Ideally in video (subjective I know), the shutter speed will remain the same for the duration of the recorded clip. And for the final rendered movie it will be constant throughout. This is based on the film world's 180 degree shutter angle rule which would exposure the video frame for 1/2 of the time it takes to display it. Example: 30frames per second would use 1/60th of a second for exposure. This is mainly to give a generally acceptable level of motion blur in each frame. If the shutter speed is too fast it will start to look choppy (eg Saving Private Ryan). There are cases where the shutter speed is set considerably faster - think slow motion - this is to give tools that generate artificial frames between real frames very sharp input frames (eg Twixtor).
The other reason for full manual mode (fixing shutter, aperature and iso) is that for the duration of the clip you don't see any ratcheting of the parameters being adjusted by the camera. I think dedicated video cameras do this much more smoothly. DSLR's are rather abrupt. Perhaps the LX7 will do this more visually friendly like the camcorders do? Its easy to test, just record a clip with the shutter speed locked and pan the camera from bright to dark and play back and see if its acceptable to you. On my DSLR, the ISO stepping does not look very pleasing at all.
In bright clear ocean water (eg Cozumel) you can probably set f8, ISO 100 and shutter speed of 1/60th (for 30p) and get great clean video. As soon as you go deep or late afternoon you will start opening up the aperture or increasing the iso (before the start of recording) to get proper exposure. Then do this until the video is just too noisy (from high ISO) or the depth of field is too shallow (from the aperture being wide open).
If the video is just for fun for you and others, then most probably won't even notice most of this stuff. On the other hand, if you are getting more particular about the video and you see the result of the camera automatically stepping ISO or aperture during a clip will probably start to bug you. People seem to love GoPro video and it is automatically changing these parameters all the time. But larger photo based video recorders may not be smooth enough.
Definitely do manual white balancing if possible. The camera will probably do a pretty good job of this to about 60ft or so. After that there is so little red or green left anyways. I prefer camera based manual white balance vs the red filter unless the red filter can be flipped in and out easily underwater.
Oh yeah, after my last trip I was shooting 24p (shutter at 1/47th). I came away not so happy with the look of some of the video. Sure, it was captured to "cinematic" guidelines ... but fish can be quick and when they are moving quick the longer exposures just didn't look that good to me. So, perhaps others will chime in on whether 30p is better for underwater footage or not? That being said, wide angle 24p of an entire scene underwater is very enjoyable to edit, watch, etc
On last thing, you may want to shoot more of the video at f8 (or even a bit more stopped down). The depth of field will be rather forgiving and thus less likely to have video clips where you can see the camera trying to autofocus. Once again, perhaps the LX7 can focus rapidly in video mode in which case it would not be as much of an issue.
Seems like a sweet camera!!!
