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A neon bulb across the cord will not draw much current (with a proper limiting resistor).
I'd personally use a incandescent and a small transformer (I have one from a BT TV ant amp PS that is smaller than any AA5 speaker transformer that would be great for this application).
Another option is to lookup the series heater current of the tubes in it, and get a 1-8V bulb that matches that current*, and put it in series with the heaters. You can often get away with adding as much as a 12V drop to a series string of tubes....ESPECIALLY if your line voltage is 125V or more...Many sets were designed to run ideally at 109-117V, and could stand an additional 10-20% change in line voltage about the ideal design point...
*If you can't find one, get a bulb with less current and calculate the bulb's hot resistance using Ohm's law and the current/voltage specs of the bulb, then grind the same equation substituting the bulb current for the heater string series current (that gives you the resistance you need), then calculate the parallel resistance you need (ask google if you don't know the formulas).
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