What you need is a ceiling medallion. That's a fancy name for "a circular thing you stick on the ceiling to look nice and cover a hole". A lot of them are designed to actually have a lamp hang from them, but there are some with a big hole in the middle which is exactly what you need here. There are simple ones:
and fancier ones:
Some are white or metallic ...
You get a usb charger, and wire up the lamp in 5V, going to LEDs. (There are 5V LED bulbs, or you could hardwire them in, depending on the type of lamp you're building.) There is no ground wire, but it's low voltage, so there's no safety issue.
EDIT: 12V is also good; you can use whichever is most convenient.
If it was me I would go to to the electrical supply store and purchase one of this type of safety GND wire pigtail.
Loosen the screw on the lamp mounting bracket and install the spade lug on the pigtail under this same screw. Re-tighten.
Cut off the end with the green screw and strip the wire insulation about 3/4 inch.
Remove the wire nut on the GND wires ...
That looks like a torpedo bulb with a candelabra base (the base indicates globe style, it's mislabeled). If the voltage is at least accurate, that 29w should be a 40W incandescent equivalent, which is how many manufacturers often indicate wattage on packaging. They also make LED versions of virtually every bulb nowadays if you wanted to get an even more ...
As suspected by Ecnerwal and JACK, pulling down with moderate force eventually released the whole fitting, which was being held in place with sprung fins. It did result in a bit of the ceiling plaster coming down as dust in the process, but it seems to be the way it is expected to work. Pushing back up caused it to latch back in place.
Why this often doesn't work
The #1 reason this fails is that the ballas-- hold on. The #1 reason is that Feit Electric is a bunch of junk. Along with Utilitech and Lights of America, these are brands to be avoided. All of them use the "Chinese dumping" business model of disgorge staggering quantities of hastily built lights of cheap parts with little QA - ...
If you don't think that a plain color medallion will look good, I would consider a thin (~1/8 - ~3/16) stainless steel disk about 7" inches in diameter (not too expensive on ebay). Cut three holes to match. Polish with brasso and/or use an appropriate grit sandpaper to match the finish on your canopy.
Hopefully, the holes can be fairly small. If you need to ...
shade ring or lampholder skirt. They come in different sizes. If your lampholder is that style (with a low energy lamp) you'll need an E27 shade ring and a 40mm hole in the lampholder.
Ordinary standard Bayonet Cap lampholders are smaller diameter.
At least in the US, it is normal for lamps to not have ground wires. As long as there are no exposed metal parts, there's really no reason to ground it.
If there are exposed metal parts, you need to insulate them from either the electrical wiring or the user somehow.
Chances are that it's the bulb, and not the ballast. It takes two good bulbs to get light. Swap bulbs around see if this eliminates the problem.
Bulbs are fairly cheap. You don't need fancy ones, just your basic 10,000 hour one.
Start with getting two new one bulbs. Put this pair in each fixture in turn. Write "B" on the white metal ballast cover if ...