Compute RMS value of non-stationary signal
The general way to obtain high resolution envelope of a signal is called envelogram which is the magnitude of the analytic signal obtained from your original signal(the analytic signal obtained by using Hilbert transform).

Simply take FFT of whole signal to obtain the spectrum, then replace the value of spectrum over negative frequencies with zero (keep the value of spectrum over positive frequencies as before). Now taking inverse FFT will give you a complex signal which its magnitude is the envelope of your signal which is the instantaneous power or RMS.
By applying a band pass filter around your fundamental frequency you could obtain the RMS of your fundamental frequency.

Another way is to use time-frequency analysis which is similiar to the method you suggested (it uses a moving window instead of splitting signal)