Found myself putting TinMan's video on 'Fullscreen' and measuring against the monitor screen for the tower height lol
Thanks for the encouragement, it's a quirky setup and quite fascinating.
A test circuit was built last night and today, to see how it would run, as a solid state Bedini device. The idea was to first see if it would run and secondly, to use low power as a reference to the full 12V seen in TinMan's video.
I haven't found the specific wirings of his circuit yet and was worried too that I might quickly fry the LED..
Having never built a solid state Bedini circuit, the one that Minoly posted recently on Energetic Forum was used as a reference. However, that circuit seemed to suffer from updated components and swaps and changes within the thread, which left me unsure as to the specifics again with that one. In the end, I went with a hybrid of components and wiring, similar to a usual magnet wheel based setup. All needs work, all is in a state of flux, as it were.
Immediately apparent, on firing up with a li-ion cell, was that the pancake needed flipping - the tower is wound counter clockwise, so the pancake needed to be clockwise...it's just how these Tesla setups roll.
Then, wahay, the neon lit up
Charging is possible to a 12V battery !
Also, the circuit will run and the LED light up, right down to using a 1.2V Ni-MH.
The down side is the current at the moment, which is about 170-200mA when using the li-ion...so it's nowhere near the 1mA figure yet.
But, i'm not using a 12V source yet. Mind you, Bedini says you should be drawing 2A when charging a 12V car battery, so I guess it's not really a bad circuit. Will desolder that one and move toward 12V and a better circuit replication.
Quick video, showing a few tests:
http://youtu.be/wWOCC1FW968*Update - found an MJE3055 (newer plastic package version of the transistor in the TinMan video). Soldered it up and...nothing

Looks like it's blown. So, i'm back to the C4242. These have a healthy 400V rating and yet fire right down to AA battery specs, so they are good for this

If I get favourable results tomorrow, then an MJE13009 and more will be tried, to ascertain whether a particular transistor displays different low current effects than another.
*Update 2 - Back to the C4242, but otherwise exactly what I understand to be a normal SSG circuit, with trigger to Positive rail, rather than Negative rail as described for the magnet wheel devices. Reviewing the video and related discussion thread, the resistor was changed to 10 ohms.
I have the green LED rather than a diode across the Base and Emitter, otherwise it's as TinMan has his and the function is the same, to limit the voltage across the Base.
A 150K pot is used, with his being unknown.
Power is from a 12V 4A regulated switching supply that i've cleaned up. It seems to be a 1970's model, still runs well though

A test was run just now and it used 230mA on the input, the pot was twizzled and for a brief moment, with meter set to the 2mA scale, my meter read 1.157mA. However, the neon wouldn't light at that low an amperage

The neon lights fine, when the circuit is running much nearer to the 230mA.
My scope shows a sawtooth - though I have a ghosting above the image and so the scope may need attention.
Removing the battery connection shows a rippled sine wave and that's been a bug in operations that I haven't worked out...it's mains hum coming through I believe, because touching the probe lightly with my finger produces something similar.
A video is unfortunately not possible of this setup...on returning the pot to a level that would light the neon, my meter went flying past the 200mA draw level and a small 'pop' was heard from within. Sounded like the protection fuse blew and I hope it's got one fitted

Thoughts
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TinMan appears to have made an error in his device hookups. KyleCarrington mentions on the YouTube comments, that the output voltage reading is dependent on the battery actual charge level and, we're not seeing anything other than that. Tinman is correct that 1mA will be so light that the resting voltage will be shown, but that's not the voltage he's creating and going in from the system, it's, the resting voltage of the battery !
Caveat, the arrangement he has is definitely better than mine and so is his test gear, so the 1mA neon lighting is still up for debate.
All in all, the pancake and tower DO create a Bedini trigger and a waveform is shown on the scope, denoting charge output. It has shown to be a great step up transformer too. However, the issues remain of the output figures and connections that TinMan shows in his video.
Pic below of the closer replication: