I'm going to put my naive inductor investigation on here so that hopefully the comments will come alive with illuminating advice.

Firstly, I found this website which tells you how to use the "ring down" method to evaluate the losses of inductors. Great. It says the rise/fall time of a 555 timer is too slow to drive the "hitting" of the "bell" that is the inductor. Quick search say's that's 100 ns. How fast is my ATMEGA328P based Arduino or my ESP32? Oscilloscope to the rescue:

Here's the Arduino running at 5V - rise time is  4.5 - 4.8 ns

And here's the ESP32 running at 3.3V - rise time is 2.8 - 4 ns

I think we'll stick with the Arduino for now, but we are roughly looking at the same gradient, just different voltage levels.

So my first setup used the "weak capacitive coupling" method for driving the inductor (see website link above) and looked like this (driven by pulses from the yellow wire on the left):

Which resulted in this measurement on the 'scope:

For the ring down method, we're looking for how many oscillations happen before the amplitude halves. In this case, with a 7.2mH TDK inductor I got. It's not looking good, so like an exasperated parent thrusting their child towards a godparent to "talk some sense into them" I asked the eponymous Ted Yapo what might be going on and he said my scope could be stealing some of the energy it's supposed to be measuring. 

So I added a 9.1 MOhm resistor in series with my 10x probe in an attempt to slightly improve things. Here's the lash-up rig with the TDK inductor attached at the bottom. You can't see the parallel capacitor but it is there!

Here are the inductors I was interested in:

L-R: TDK 7.2 mH, Bourns https://uk.rs-online.com/web/p/wire-wound-surface-mount-inductors/8118777/ 2.2mH (which is what was in the V1 #Yapolamp ), a range of Coilcraft: 3.3mH, 6.8mH and 10mH.

Here are the respective results shown on the 'scope with nothing changed:

TDK:

Bourns ( did zoom around but couldn't find any oscillations that looked like it was ringing):

Coilcraft 3.3mH:

Coilcraft 6.8mH:

Coilcraft 10mH:

Suffice to say, I probably shouldn't have used that Bourns inductor in the V1 Yapolamp. It will be interesting to see where the failure of that version may be found to lie. I also have a different energy plan in mind for the V2, so there would potentially be more budget (as if I even have a budget!) available in the BoM for a nicer inductor.

Or maybe, where we're going, we don't need inductors...