"One person’s suggestion for reproducing the Rockmore sound is on this page: http://aetherwellen-musik.de/Theremin-Instrumente-Instruments/Claravox-Centennial-CV-C-Theremin-Review-Improvement" - bendra
Man that guy is dedicated!
From the gift tape I can see the 2nd harmonic is rather high, and the 4th harmonic is generally suppressed. Some suppression goes on somewhere between the 7th and the 8th too. Low notes have an almost flat spectra, lots of harmonic content, which you can of course hear. Lots of the low end timbre is audibly coming from the low damped speaker cone just flopping around in its magnetic gap. To come close to the room recordings would probably require something as complex as a guitar amp & cabinet simulator.