Ysiegel, the Baal Hatanya’s innovation in mikvaos had nothing whatsoever to do with Shabbos. His two big innovations, of which he was justly proud, were the use of sharp but fragile knives for shechita, and a contraption to enable WOMEN to tovel in warm water.
There were no rainwater mikvaos in his day; therefore there was no such thing as a heated mikveh, and certainly not on Shabbos.
also, you have completely missed the point about the filter. If the filter is on the mikveh is unfit for use. Therefore using it on Shabbos would be a problem.