Thanks for the detailed reply! I'm not going to send them back for a refund because the lights themselves work for what I want and I've already drilled a whole lot of large holes in the bottom of my new riser shelf to accommodate them. My solution will be a low-noise constant current source for all of them based on loose junk I have laying around. However, this has piqued my interest for a few reasons--this would be an opportunity to learn something detailed about EMI output issues and also I'm a bit alarmed by the effect it had on devices that I didn't expect would react that way. In theory I understand how HF EMI can affect a DC circuit--rectification, just like an AM detector--but not knowing how, where and why this happens on a microneolithic device like the 731B makes me worry that some less outrageous EMI issue could cause a smaller error, one that wouldn't immediately yell 'broken!'.
It's certainly a textbook example!
Sorry, I omitted a photo of the other side. There is a small capacitor on the hidden side of the transformer, right near the LED output connections. I assume this circuit is similar to a flyback design?
Ah, then that explains that. Yes, it sounds like a flyback, so, primary winding, some ratio, secondary, and the duty cycle is more modest than a direct buck would. Should be able to identify similar circuits from western manufacturers; although, at a glance, I don't see much quite like that from Power Integrations, they've advanced a bit I think (e.g. regulators with integrated isolated feedback, or secondary side synchronous rectification -- sweet!).
OK, so if I look at #24, the primary turns on at the trigger point, then off about 3us later, the secondary conducts for ~10us and then when that cuts off the transformer rings until the next primary turn on.
Yes, exactly.

I took most of the captures with the 20MHz BW limiter on, but at the end I realized that was a mistake--there's plenty of energy above that. And it seems to travel right through almost everything in its path, including the in-guard isolation of at least one old DMM.
Aha; yes, it'll tend not to "stay in wires" at those frequencies. Much to the chagrin of your other equipment.
As to how they're susceptible, dunno; at worst, it could be mere gaps in the housing, but more likely it's picked up on wires/cables hanging off it. I'm not sure exactly how "wireless" you had those things; battery power at least suggests no conducted mains EMI, but an attached cord could still be an antenna, and probes/cables can be as well.
You'd hope that they're more immune to such things regardless, but eh, it depends. A lot of classic BJT-based hardware is susceptible by RF rectification at junctions; the inputs could be better filtered (or indeed, the mains or any other means of RF ingress), but maybe it wasn't important at the time, or may not even be feasible (when the input bandwidth and impedance preclude meaningful RFI filtering). That includes BJT-based ICs, of course. Which nowadays, often come with RFI filtering on chip (e.g. OPA192), but also, you can't do that so well when you need 100s of MHz of GBW.
Yes, the measurement was quick and primitive, now I'll actually have to think about a homemade LISN and the LISN-mate device from another thread to separate out CM and DM. Or, if you have a suggestion how to better do it quick and cheap... When I get a bit of time--after I've scraped out the 9 LED drivers that I glued in place and replaced them with a single CC driver--I'll disassemble one and draw out it's circuit. I can then try your various suggestions to mitigate the noise. I'm assuming the spikes are the EMI component that is actually causing the issues I see. I'd also be interested to see what can be done to effectively filter those types of EMI out other than modification of the source.
Don't worry too much about a precise well-specified LISN -- any kind of impedance stabilization will help.
First, get everything on a ground plane so you have something to work against.
Second, get your isolation transformer down there: maybe some bypass caps (a couple uF from each line to GND, maybe with some resistance to keep it from ringing with the transformer's capacitance -- 4.7 ohms is typical), maybe a line filter (to improve immunity from mains noise), etc.
Third, the impedance stabilization itself: you want an R || L characteristic, so the load sees fixed R at HF, and L at LF (in particular, the impedance at mains frequency will still be quite low). The R could be straight in parallel with the L, but that wouldn't be very helpful; instead, we wire it to GND, and avoid burning mains voltage by using a series coupling cap. Furthermore, we can use a BNC instead, and hook up whatever radio receiver we like (scope, spec an..) as the resistance. Or a terminator if unused.
So, if nothing else, you could assume the iso trans is some (well enough behaved) source impedance. In general it's going to have some resonances in the 100s kHz to MHz, some weird combinations of transmission line effects due to the layers of windings -- and that's probably not going to make a good LISN, so you'd want to test it to be sure. But as a hand-wave, I mean, that's basically what you've done already, and as you can see, it's hardly a game changer, you can still see the thing is blasting out the noise!
And so we get typical values like the 50uH for a FCC Part 15 or CISPR 11 sort of network, maybe some bypass cap on the mains side, and a ~0.2uF coupling cap to the 50 ohm receiver.
Then the EUT can be elevated modestly above the ground plane. In a test lab, it might be on an insulated table, and you're just measuring any and all noise in the chamber itself; for a more compact environment, you might prefer keeping that noise near the ground plane (so, a few inches elevation say), where it's less prone to picking up ambient radiation / losing energy to the ambient. Note this gives impedances in the 150 ohm range for cables elevated over the ground plane, handy for calculating impedances/transformations -- if you're, y'know, into that sort of thing.
And then the whole thing is floating, so you won't need any other connections, no additional LISN or CDN or whatever. You can use one on the LED side if you like, to see what's going between LED and mains cables, (somewhat) independent of frequency (since the capacitance of the EUT over ground will be the only CM "reaction mass", and thus give an LF cutoff). Should also be okay to ground one side of the LED, to the same end but with slightly different gain, of course inviting some possible reflections at much higher frequency.
Anyway, dealing with the noise at the source is of course ideal, but if you can't do that for [reasons], then consider putting the module in a metal box and using pass-thru filters on all connections. Maybe one can be hard grounded (LED neg?) and the rest get whatever appropriate filtering. So, other LED terminal, probably a small inductor and relatively large cap (0.1uF or more?). Mains, a CMC, with X1 caps between lines, on both sides, and Y1 cap(s) from one or both lines to the enclosure, will do.
I tend to leave off pairs of Y1 caps, but eh, it depends. I mean you often see a Y cap from each line to GND. It's not very important, most of the time, because the X cap has a low impedance: the two lines already act together at most frequencies, so the second Y cap is basically acting in parallel. Maybe it helps to improve balance (so, CMRR), maybe it's not needed.
For better sake of discussion, consider:

"AC line" is the LISN equivalent at pass frequencies (so, 100s kHz to 10s MHz).
By paired Y caps, I mean you might have one from neutral to GND like C4, and one from hot to GND as well. C1 is much larger than C4 or the other one (absent here), so L/N tend to act in parallel. But you can see they act as a balanced divider with respect to DM noise, so that can be a reason to use two.
C3 represents isolation (inter-winding) capacitance, in a typical DC supply. That might not apply here (i.e. if secondary is grounded to primary), or maybe it does. We most often place C7, to shunt this at the source, i.e. between pri/sec common nodes, greatly reducing the amount of filtering needed around it.
The matter of C5 vs. C7 vs. C6 (note the optional GND by C6; in this case neither the secondary side GND nor C6 would be present, as your LEDs are completely floating*) is often down to experimentation; it depends on exact circuit topology, and like, winding details, so, is hard to anticipate or model, and easier to just provide the placement options and try them when EMI testing.
*Actually let me modify that a bit; we can model the CM impedance (the capacitance of the LEDs and wiring to the surrounding space) as some impedance like C6, though it's going to be a more complicated equivalent circuit once you include all the wiring and stuff, and radiation resistance. So, maybe it's only like 100pF, not the ~nF of an intentional filter cap. And that's where the "reaction mass" comes from that allows noise current to be backfed to the mains side.
And note that the CMC has a lot of leakage (typically), so the DM filtering between it and the X caps can be quite reasonable. So that often takes care of CM and DM at the same time.
What you can't do, is to put in
just a mains filter: if it has Y caps to GND on the load side, well those just bypass all your noise straight to mains/ground and your LEDs are still noisy as hell. So watch out for that. The kind with a CMC facing load should be okay/helpful, but the impedance of that CMC will still provide some reaction against which to radiate via the LED wiring.
If you can get some CM impedance to GND on the LED side, then the filter has something to work against, but watch out that the loop between filter, module and GND return path (back to filter GND), is short -- otherwise it'll act as some degree of antenna itself, too. All of which are avoided by cramming the thing in a box... hence that option.
And, it doesn't need to be a
box box, you can open a hole in one side for example, if there's nothing immediately around the hole; even open it up so far that there's little more than a ground plane under the module, with some filters on the connections before the wiring connects. The more open it gets, the more chance there is for induction/radiation, and the more important it is that the circuit sits close to that plane, and the less safe our assumptions are about the box-like-iness shielding situation. (Planes do work quite well, which is why so many things can get away with just being a PCB in a plastic enclosure.)
The important part is eliminating AC voltage drop between wires (as in vs. any given wire (normal mode), or within pairs (diff mode), or between pairs (common mode)). Having all the wires group together and get filtered in one common location, minimizes that voltage drop between them due to any other stray induction or voltage drops or whatever. (It's often better to have input and output on a common side of a supply, so that they can be filtered against each other locally, without having to also filter against voltage drops along the length of the board.)
Tim