Thanks, you put in a lot of work!
I would disregard the noise data of the UV-LED, as the Vf, which like the blue, white, some of the green LEDs, gives a much higher than the bias voltage needed for the circuit. Your notes suggest that 681 Ohm Rk gives about 1V so should also any diode.
With a UV-LED the bias voltage get up to around 3V and and raises the plate voltage almost 100% and put the tube close to the B+, so a non-starter to use anything that puts the plate voltage away from design center, as you would lose dynamic range. OK maybe for measuring noise signals, but large signals would clip.
Diode bias with its low dynamic impedance would kill cathode degeneration and increase gain, versus an un-bypassed cathode resistor.
To evaluate an LED for the application it should have the desired Vf, in this case an IR LED could work, check the data sheets for suitable types. Digikey has good filters for selecting parts, including "Vf" for LEDS. Your circuits looks like it draws about 1mA thru the plate. The LTE-4208 looks like it could work, like 1.1V at 1mA, data a bit fuzzy at the bottom, may perhaps need a another mA to get it started in its linear range, where according to the slope of the curve would have an impedance of ~7Ohm providing very little negative feedback. This could be bypassed with a "big" electrolytic, but as the slope is very linear, whatever microvolts of feedback would have very little effect and can be disregarded. The original Neumann circuit ran the heater current (40mA) thru the cathode resistor thus reducing plate current (0.55mA) influence 38dB, making the need for a bypass capacitor small.
Using 1A, 1kV, fast switching diodes in this position is not necessary for a 1mA, 1V, DC only circuit, but wont hurt either.
If light inside the enclosure is an issue, an LED can be covered up with some black heathshrink.
I ran the noise test with two UF4007s in lieu of a IR-LED in a 45mA/V very low noise tube (gain about 80) used for moving coil amplification, which draws 20mA from a 185V source, (not necessarily a top choice inside a U47 body unless some neat heat sinking is used, 3W vs. 5.7W ).
I could not measure any advantage of the UF4007 pair with the 6.5 digit 300KHz DMM. My tube is fed by a constant current source which additionally suppresses any current variation in the tube. The current noise in the diode is converted to voltage over the diodes dynamic impedance in the tube circuit.
Signal to noise is important, is your noise measurement below the noise in a quiet room? Any signal would bury the noise and only be revealed during no signal passages.
Very few of Fluke's handheld electrician grade DMM's have enough bandwidth to cover the audible range. The old and very slow Fluke benchtop 8050A does cover it up to 100KHz. Most DMM's give up at 1KHz. The top end handheld 87V MAX DMM barely makes 20KHz at below 0.6V.
While a diode junction will generate broadband noise also in the forward biased direction, not much of it is relevant to the audio spectrum in this application, quoting Mr. JR.
https://groupdiy.com/threads/do-leds-generate-audible-noise.82153/
Walt Jung's
Directory listing <-- GLED431 ~1nV/sqrt Hz
Using diode(s) looks like you got 2dB lower noise over (unbypassed?) 681 Ohm resistor.