Alternate driver circuit idea.
Some solid state Neon light power supplies use rectified 120VAC to get 200vdc which can directly drive a 600VDC 555 equivalent chip: IR2153-S. This chip has dual outputs that are designed to drive dual IRF640 FETs in a push-pull configuration to minimize the rise and fall times and generate the sharpest possible transformer primary voltage swings. The more intense the primary swing is, the higher the secondary voltage will be. This results in consistent high frequency / high K-voltages.
I almost finished the schematic of an Allanson Neon Supply SS935ICH (attached). This was a potted device (circuit board was encased in epoxy resin). They are about 40-50 bucks and run at about 22kz. They contains a dual shutdown safety circuit, they will shut down if the load is too high, or if the load is too low. So it made it frustrating to test on a Cell.
I figured if I could double the frequency, remove the safety and run it at 1/10th of its power level I could see how it behaved with a Cell. Never got around to it tho..
Chewie.