What about switch at 20kHz and use resonant tanks to smooth it into a sine wave?
You would need an impractically high Q resonant circuit to handle the grossly non-linear and variable load without severe distortion. Then there is the design issue of coupling to the resonant filter(s) and the attendant problem of amplitude stabilisation.
A “signal” to switching ratio of only 5:1 isn’t really an issue here – adequate “carrier” suppression can be had with more low-pass filter poles. What makes it all impractical is the incompatibility of any practical filter with such a non-linear load.