Windowing of signal before FFT
So you think that the energy of the windowed signal in the frequency domain equals to the original signal multiplied by a certain value related to RMS of the window function, which is wrong. According to Parseval's theorem, the total energy of a signal can be calculated by summing power-per-sample across time or spectral power across frequency:
$$
\sum_{n=0}^{N-1} |x[n]|^2 = \frac{1}{N}\sum_{k=0}^{N-1}|X[k]|^2
$$
Hence I'm going to prove your statement wrong in the time domain.
The energy of a windowed signal $w[n]x[n]$ is
$$
\sum_{n=0}^{N-1} |w[n]x[n]|^2 = \sum_{n=0}^{N-1} |w[n]|^2|x[n]|^2 \neq C\sum_{n=0}^{N-1} |x[n]|^2
$$
where $C$ is a constant. The above inequality holds true as long as the window function $w[n]$ is not a rectangular window.