I worked on the OSEM box a little more today, with the hopes of reducing the measured output current noise. I succeeded, at least modestly. It turns out that most of the noise was indeed caused by the crappy resistors.
Below is the circuit for one of the 5 LEDs. The output of the op-amp structure directly after the LT1031 reference is split between 5 stages identical to the structure on the right. I have shown just one (UR) for clarity. The various measurement points are explained below.
I started from the beginning of the circuit, directly after the LT1031, to make sure that the excess noise seen the other day wasn't just from a noisy reference. Below is the measured output voltage noise along with the LISO estimate. Clearly, the LT1031 is performing to spec (as it should, since it's a new part that I just put in). Note that the apparent better-than-spec performance at low frequencies is just from the AC coupling, which I needed due to the high DC level.
Since the reference was in order, the next step was to switch out some of the crappy old resistors for nicer thin-film ones. In case anyone is interested, Frank has done some detailed investigation of excess 1/f current noise in resistors. I measured the voltage noise level at the point labeled "inter-stage measurement" above, first without any modifications and then after swapping the old 10k resistors (R1 & R2) out for nice Vishay thin-film ones. There is clearly a big improvement, and the modified circuit essentially agrees with LISO now down to 1 Hz. Below this, it looks like there could still be an issue.
I wanted to see what the improvement was in the overall output current noise of the system, so I went about measuring the current noise as I had the other day (by measuring the voltage noise across R55 and dividing by the resistance). The performance was already better than the old measurement, but not at the LISO level. So, I replaced the current-setting resistors (R54 & R55)---which were actually 3 parallel resistors on a single pad in each case---by nice Vishay ones, as well. I didn't have any that were close to the original resistance of ~287 ohms, so I put three 1k ones in parallel. This of course shifts the resistance up to 333 ohms, but that only causes a ~16% change in current. I was sure to convert voltage noise into current noise with this new resistance, though.
With this change, the total output current noise is now very close to the LISO estimate as well down to ~1 Hz.