If you know the formula, you could integrate and calculate its sqrt analytically, then apply the obtained formula on the MCU.

I do not think that you have thought that through. You have certainly not explained how it might help.

The thing you would need to integrate would be the square of the result of applying the transfer function to the discrete time series sample data. You can not helpfully analyse this before you know what the data are.

The only case where the integral of the (square of the) transfer function alone could be useful is if the samples happened to form a straight line. But then they do not cover an entire period of the waveform, and the RMS value is of no significance.

LMGTFY... RMS d' WIKI, check under the discrete section and formulation. its possible i think.

You are just waving your arms.