I had to build a tube theremin to understand how the authentic classic sound comes about. The main secret is the two pitch oscillators must not know the other one exists. Separate the two oscillator coils as much as possible. I believe that was the fatal flaw in the Keppinger design, weak inductive coupling between the coils. There is nothing more that is special about the coil design, notice I use the standard IF coil. Here is my tube sound using two of the boards seen in the image. Oddly a tube theremin design has very little, if any thermal drift issues.
If you dare my final analog build with classic passionate expression: solid state sound sample.
Good Luck with your build as she will test you, over & over. . . LOL
Edit: Looking at the schematic L1 & L5 do not have a DC current so if all else fails you could hide the IF transformers inside the coil tube at the base, you are already faking it, the classic sound is what you want. The IF Transformers I use the coil is inside a ferrite cup which better contains the inductive field.