With no luck on my CCFL design, I decided to finish the lamp using LEDs. I bought a variety of cool-white and warm-white LEDs for experimentation, and decided they would be perfectly good for the desk lamp.
After consulting some Art Deco books at the library, I designed a new lamp shade with concentric acrylic disks around a central tube holding the LEDs. I used about 12 LEDs in several sets of 12v clusters. I made the center tube from PVC plumbing pipe and copper spray paint, keeping the lampshade as light-weight as I could. I held the disks to the tube using old bicycle wheel spokes. The nuts from the wheel rims threaded on to the spokes made the stacking assembly very simple.
In the end, the LEDs worked great. I gave the lamp to my father for Christmas (a year late!).
I eventually found out the cause of the CCFL problem. The CCFL bulb works on very high voltage; that is why it needs the inverter board. This means that with a low wattage, the actual current flow is extremely small. So small, in fact, that just an extra 10 or 20 inches of wire is enough resistance to keep the bulb from getting the power it needs and lighting up. So I’m sure it is solvable with some combination of larger wire or a modification of the inverter board to boost the power. Perhaps on the next lamp…