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How about installing a CMOS 555 timer to drive the grid, so the first stage becomes just an amplifier? You could run it off a battery, or maybe even from a little current bled from a convenient cathode somewhere, or a large resistor and a zener off B+. You would need a trim pot to get the exact frequency, though. A more accurate option would be a higher frequency crystal and frequency divider.
Just throwing out ideas because it's easy for me to say!
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