The original lamps had upper and lower sockets and a 4-way switch so that either bulb or both bulbs could be illuminated.  I'll be using LED strips to create a single vertical "bulb" down the center of the lamp.  I COB LEDs for more uniform lighting, at 3500K for a little less yellow light than "soft white" (the fabric on the shade will likely add back some yellow).  I selected a dimmable driver that can be controlled by a potentiometer, both of which I plan to install in the base.

Here are the main components laid out for testing.   The aluminum channel length is such that it will sit around 3-4 inches below the top of the shade.  I wanted to make sure it wasn't visible by someone standing a few feet away. 

Full brightness:

Dimmed:

Strips adhered to the channel and clear shrink wrap applied to the soldered ends.  The perforated hub in the lower left will organize the channels and the wiring at the bottom-most end.  There will be a similar channel organizer in the middle and the top (without any wiring holes).

Here's the assembled "bulb", currently held together with some rubber bands.  I may or may not use the channel diffusers.   If the light is uniform enough through the shade without them, I'll leave them off.

Here's how it will sit in the lamp.  I'll be able to adjust the height with a nut on the lamp pipe all-thread that the lower plywood organizer disk rests on.  The all-thread will ultimately be sleeved in a brass tube where the all-thread is exposed.  I'm still trying to source that.

Testing it out.  I think this will work fine.

With the red rosin paper shade, this would be a good bordello lamp.