Alright, it appears I've gotten to the bottom of the shakes –at least for now.

I pulled the hardlines and removed each valve assembly for inspection and cleaning.
What I found within the assembly was interesting. Unfortunately, the environment was dark and difficult to photograph. There was a bit of fine particulate in bores 2-4, but nothing too bad(?). I was able to clean them with q-tips, followed by a thorough priming.
Bore one however contained quite a bit of reddish, I'm assuming rust. That took a bit of swabbing but was eventually returned to a clean state. Injector 1 had been nailing a bit (confirmed by loosening hardline) during cold starts.

The valve holders were all a bit rusty/reddish, though valve holder 1 had the most of this stuff on the outer surface. I have no idea from where or how it came to form.

The surface of each plunger was fairly smooth. No varnish or crud. There was a thin line however that marked the spot the plunger made contact with the valve holder. I then employed Diseasel's cerium oxide method for polishing the bevelled edge of each plunger and holder. I polished until the line stopped catching on my finger nail, though I was never able to remove it entirely.

Here's a photo of each assembly (the detail on the plungers was impossible to capture).



As you can see, number three's body is very scored, I assume, from spinning within the bore.
One and two aren't terrible and four seems okay, though my assessment is arbitrary.

I reinstalled each DV along with new valve springs and crush washers.
So far so good. The side to side oscillation is gone.
It seems the issue did in fact lie somewhere within the Injection pump, though too many variables were affected to pin anything to a single cause.

Has anyone else tried this? What did you find?

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1981 240D 4sp manual. Ivory White.