Hook your AC voltmeter across the dial light with the set off. When it first comes on, at what does the voltage first peak? Then it settles down to 8.1VAC with the bulb lit, right? If the peak isn't too much, you'll just need a simple resistor in the bulb line to drop the voltage to 6.3V or a tad under. About 15 ohms at 1 watt.
Your switch contacts are dirty or have lost their springiness and aren't making good contact or have a balky snap mechanism. It's possible to drill out the rivets and take it apart and fix it, but it's a lot easier to just replace it. Unsolder the wire from the high end of the pot (terminal farthest from the chassis ground terminal.) Measure the resistance between the two end pot terminals. It's probably 500K or 1 megohm. You'll need a pot with audio taper and SPST switch attached. The length and type of shaft (round, half round, etc.) have to be noted. One place to get them is from Mark Oppat. He can fix an old control or provide a new one.
Edit: actually, looking at the picture, if that pot were taken out of the chassis, you'd bend the tabs up on the front of the pot and separate the pot from the switch. Then looking inside the metal shell you'd see how to get to the guts of the switch.
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