For the sake of completeness, I'm reporting back after finishing the monitor upgrade mentioned above.
Unfortunately, the results so far are mixed. The larger screen (10.1" v. 7" diagonal) of the Uperfect 10.1" 2K monitor (available now on eBay for $89.95, which is $15.00 less than I paid when I ordered it 10 days ago ... grrr) is great and the additional resolution (2560x1600 v. 1920x1200) is definitely useful when reviewing captured photos before I strike my setup and move onto the next scene / photograph.
But the quality of the image displayed isn't as good as it was using the previous monitor (a 7" Aputure VS-1 FineHD), which, of course, cost roughly twice as much, so I suppose this shouldn't have come as a surprise. I'm not an expert in these matters, but the issues seem to be that the Uperfect monitor has neither the brightness nor contrast range to match the Aputure monitor, let alone improve upon it. To make it usable for my nighttime photography, I have had to crank the individual RGB channels as high as they can go while preserving the same color balance, as well as crank the overall brightness control to its max. Even then, it doesn't reveal sufficient detail in the shadow areas and the highlights are nearly blown out, which makes judging exposure lengths a bit tricky, because until I become accustomed to this monitor's performance, I don't have a very good idea of how the exposure will ultimately look (except to check the histogram, which isn't very precise because it's based on a .jpg and not the raw file) or how much headroom, if any, I will have available during post-processing.
This also means I have to use another workaround as well, which is to increase the ISO three or four stops, roughly compose my photo, and then iterate to the final composition by taking test photos and looking at the results, then blindly adjust the camera this way or that. While this approach does work -- I've had to use it with many other camera setups over the past decade -- it's a PITA, to be honest, and I would be much happier if I could avoid it, because taking multiple long-exposure photos in a row heats the camera's sensor and potentially increases the noise level, especially when the ambient temp is high, as is generally the case during the hot summer months here in Phoenix, AZ.
So, Yes, this monitor setup does work -- but only just okay, sort of -- and I'm pondering where to go from here. Unfortunately, the other 10.1" monitors with 2k resolution aren't exactly thick on the ground, so there aren't a lot of other options, and of those that do exist, many appear to be the same monitor, only repackaged slightly. Or they use the same IPS panel, which suggests they will perform similarly.
However, if one is prepared to accept a larger size -- say, between 12.5" and 13.3", and spend more money -- between $300 and $600 -- then some other, presumably better-performing monitors are available, such as the Lilliput A12, which offers 4k (3840x2160) resolution! Of course, I will only be able to take advantage of that during playback, because the live HDMI signal my A7R outputs is limited to just 1080p (1900 x 1080) resolution, but it does output a 4k signal during playback, which should provide a significantly higher-res image to review.
I hate to spend the additional money and I don't really want / need an even larger monitor, but after even just two nights of working with a 10.1" monitor -- even one that is as much as a PITA to use as this one has proved to be -- going back to a modest 7" monitor and being happy will be very, very difficult for me. (And for the record, using the OEM 3" LCD on the back of the camera is now for emergency / backup use only ... no way, no how can I ever willingly limit myself to working with a screen that small! <shrugs>
Anyway, here's what the larger monitor setup looks like (note: the cables / routing still need to be sorted, so the final, final result will look much tidier, but there's no point in going to that expense and effort at this point, when I've not yet made a final decision on the monitor):

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