Rob, I looked at your blue island DNG file. The original is fine, not a thing wrong with it that I can see. I diagnose the green banding as follows. The situation you're running into is being triggered by a couple of things. First is that the image is very monochromatic in a particular range of blue hues that are very sensitive to contrast changes. When you're working on an RGB file, doing things that involve contrast changes will push and pull the R, G and B colour channels around. They don't typically change uniformly; the underlying math is trying to maintain a consistent perceptual "look", but sometimes when the colour channels move, they hit inflection points with each other where a hue shift can be caused. This is happening to this file.
Making curves edits in Photoshop to approximate the tonal appearance of your web site JPEG of this image, I can easily create green bands in the sky in the same area you pointed out. That's because by default, Photoshop curves (I believe like most Lightroom contrast edits such as the grad tool) work in colour mode, changing the R, G and B channels differently and producing hue shifts. In Photoshop, you can change the blending mode of a curves layer from "Normal" to "Luminosity", which attempts to restrict the tone curve to impacting only relative brightness of the image. But this only partially restricts the green banding from happening, because fundamentally in an RGB colour image when you impact contrast you also impact colour. The two are linked.
The second thing is then triggered when you convert to sRGB for posting on the web. Recall that the gamut of sRGB is more limited than Adobe RGB, and much more limited than ProPhoto RGB (a variation of which is being used internally within Lightroom while you're editing the file). It so happens that your range of blues here involve a degree of contribution from the R channel that gets absolutely crushed when you convert to sRGB. This exaggerates what was already happening from the contrast edits above, and really bakes in some channel inflection points that make dealing with the hue shifts extremely difficult after the conversion to sRGB.
I've attached a couple of screenshots here. The first shows the R, G and B histograms from your blue island DNG, where I converted to 16-bit TIFF / ProPhoto RGB with mild edits much as you described. (I use Photoshop, not LR, but that doesn't affect the underlying diagnosis.) All 3 histograms look normal, though clearly the R channel is biased to the darker tones, while the B channel is biased to the brighter tones.
The second screenshot shows the same R, G and B histograms after converting the edited file to sRGB, but no further adjustments. Note that the R channel has been fairly crushed, with a lot of clipping to black. The third screenshot shows a visualization of the R channel alone in the image, showing where the channel data has been blocked up completely to 0. I downloaded the blue island JPEG from your web site and looked at it; the R channel is even a bit more crushed, indicating you had a slightly more contrasty original working image before the conversion to sRGB did its damage.
So to recap, this situation involves a couple of different things. First is a monochromatic blue hue range that's very sensitive to contrast changes pushing the RGB colour channels into some hue shifts at certain points where the channels are hitting some ugly inflection points in the roll-off from highlights down to midtones. To avoid this first problem, I experimented with converting the image from RGB mode to Lab mode, and then making similar contrast adjustments to the L channel alone. (This requires Photoshop, I believe Lightroom has no support for Lab mode.) Doing this, I saw virtually none of the green banding effect, plus a retention of blue hues that were more true to the original colours in the DNG file. Lab mode is tailor made for this type of situation. Once done all the contrast adjustments in Lab mode, convert back to RGB mode and carry on with any other finishing work you want to do.
Note: If I continued to push the contrast quite extremely, even Lab mode edits were able to hit weird inflection points in the two colour channels of Lab, introducing some hue shifts. You're simply going to have to back off the global contrast pushes in this case. Go with a lighter and/or a more selective hand, dialing contrast in more selectively in areas rather than globally. Once the hue shifts are there, getting rid of them will be a lot harder than not triggering into them in the first place. But in Lab, doing a similar level of edits to approximate the contrast shown in your web site JPEG version of this shot, I saw no green banding.
Second issue: the conversion to sRGB is hammering the R channel of the image file. This introduces another pile of hue shifts by imbalancing the RGB colour channels due to the elimination of the R component from the upper sky and lower water. The 4th screen shot I've attached shows a softproofing view from a tool called Gamutvision, with a really useful gamut warning display.
The false colour rendering shows the areas of colours that are out of gamut when your blue island shot with my edits is converted from ProPhoto RGB to sRGB. There's simply a ton of dark blues that are out of the gamut range of sRGB; during the conversion to sRGB those pixels are going to be rendered with incorrect colours, which we know from looking at the histograms above is mainly because the R channel is getting clipped to 0 in those same regions.
The rendering of false colours in the gamut warning display corresponds to delta-E values in the horizontal range below, showing dE values from 0 to nearly 11. The rule of thumb is that dE values of 1 or less are basically unnoticeable colour changes; up to 3 may be noticeable but minor colour changes; and dE values above that are definitely more & more noticeable. (The gamut warning display in Photoshop shows out-of-gamut colours over a similar region as in this Gamutvision display, but Photoshop doesn't show by how much the colours are out of gamut -- just that they're in or out.)
So... the mere act of converting this image to sRGB triggers hue shift in a large area. The way to avoid this would be the same as the way to avoid any other out-of-gamut situation. If you really want the web JPEG file to look its best, then prior to the conversion to sRGB you'll need to custom edit the file to move the affected regions of the image away from the blue range that sRGB can't represent. You'd do this probably by increasing lightness and/or reducing saturation in those deep blues.
In my estimation, neither of these things has much to do with the 645Z. Certainly the second issue is just a standard matter of being able to capture and edit rich colours that fall outside the range of the sRGB colour space. Nothing to do with the camera, except by virtue of its ability to capture a rich colour range in the first place. Contrast edits simply push those particular colours further into a region outside what sRGB can handle.
The first issue of why this particular range of monochromatic blues triggers the green banding and so on is perhaps peripherally related to the 645Z, in the sense that the spectral response of the camera's sensor may be recording blues in a way that past cameras you've used do not. But that's not necessarily a "fault" of the camera, that's just the way it's recording these blues. Having become aware of it, there are ways you can edit for contrast and tone, without hitting the tripwire. The simplest of these involve edits in Lab mode which is designed to separate tone edits from colour edits.
Also, as a side note, the above confirms this file is a different situation from the other example DNG you posted. As I noted previously, the trigger issue with that first file was both raw image data green channels being heavily clipped straight out of the DNG. That's not an editing issue, that's an issue of over exposure followed by a challenge in how to do a channel reconstruction to compensate for the lost G channel raw data. This can be done in Photoshop, but not in Lightroom... or certainly not easily anyway.
