Update:
SOLVED!!!!1!
It's disgusting that it took this long to kill this gremlin, but at long last, I did so with extreme pleasure. Dawnguard lives!
The culprit turned out to be two lousy lines in skyrim.ini that were inserted when I installed a mod, months and months ago, that allowed free map view zooming:
uLockedObjectMapLOD=8
uLockedTerrainLOD=8
In Skyrim, map mode (entered by pressing "M"), rather than being a separate representation, is the game's actual map, seen from far above -- it's what you would eventually see if you survived getting clubbed by a giant and launched into orbit around Nirn. You can test this yourself the next time you kill a dragon by entering map mode just as you're absorbing its soul; you'll be able to see the glow of your own soul absorption while in map mode!
Now, the mod I installed some months back allowed you to zoom in as far as you liked while in map mode, almost all the way down to surface level again. This is useful if you were looking for some point of interest in the world that wasn't yet given a map marker. But I soon noticed that the mod had caused the map mode to look quite buggy; there was a lot of nausea-inducing texture flickering while scrolling around the map, and trying to scroll accurately while zoomed close turned out to be problematic. So I uninstalled the mod.
Little did I know that the mod had added the above two lines to skyrim.ini. These lines define, as fixed values, the object and terrain level of detail while in map mode. This means that no matter how far in or out you are zoomed, the level of detail will remain the same. However, this also had the horrible side effect of wrestling with other settings while in regular game mode. This resulted in a whole slew of problems while in exterior cells, including infinite loading screens after exiting houses, and hideous Z-fighting (two different textures occupying the same space and constantly battling each other to be the one on top) in distant scenery.
And crashes.
Lots of crashes.
Which brings me to my woes while in the Soul Cairn. This map is a subset of the main Skyrim map, similar to that of Blackreach, and the map mode level of detail settings in skyrim.ini were playing complete hell with it. This is why I couldn't pass between one cell and the next within the Soul Cairn without my graphics cards getting totally confused and utterly crashing the game outright. (This issue indeed also happened often while I was in Blackreach.)
A lucky bit of Googling while researching the aforementioned infinite loading screen problem revealed the solution: Simply delete those two lines. The effect was immediate: Not only were the loading screen freezes gone, but my map mode suddenly was crystal clear again (it had always been jittery even after I uninstalled the mod).
I wondered: Could this also be the cause of my Soul Cairn crashes? So I returned to an abandoned save game from when I was trying to kill the three Gate Keepers, held my breath, and gave it a spin.
It was solid as the greatest faces of granite in Yosemite. Smooth as tempered glass.
No. More. Crashes.
I completed the quest "Beyond Death" for the first time, exiting the Soul Cairn like a wrongly convicted prisoner being released from a penitentiary. LIBERATION.
So, dear readers, my advice to you: If you are having problems with crashing in your game, even with no mods and what you might think are vanilla settings, always backup your skyrim.ini and skyrimprefs.ini and allow the game to create fresh ones. An errant setting that you may have forgotten may be making all the difference!