Ooh! Remembered I have a check source!

Before going to bed, I tried out the RC pulse stretcher on the breadboard.  With a 100nF capacitor, and various resistors from 2k2 up to 4k7, I can say it had no detrimental effect on the circuit.  Whether it actually stretches the pulses will need the oscilloscope to verify.

Complete Geiger mock-up

 Then I suddenly remembered - Bob M1BBV once gave me a WW2 fob watch! A bit of a hunt later (also finding the replacement strap for my Fitbit) I had it in hand and powered up the Geiger circuit...

...oh my! This thing is HOT! At no closer than 2 inches the circuit was going crazy! With the dial glass against the tube, the rate was almost more than the LED could cope with!

So, that's a nice bit of Radium I have.