I thoughts I should mention that there is a famous psak of the Baal Hatanya (one of a number of psaks that he famously made, one being the use of polished steel knives for shechita) is the use of a heating system for a mikveh on Shabbos.
*Without going into too much detail*, it is essentially based on the source of the issur itself: gzeirat haambatyaot, the decree of the bathhouses. Since in the hanhaga of Chassidim to toivel, the actual reason for the gzeira (mentioned explicitly as part of the gzeira) is not at all relevant, therefore there is no reason to prohibit it.
This explains why in many Chassidic communities there isnin fsct heating in the mikveh on Shabbos.
Filter is not a problem here because bepashtus there is no borer (what would the ochel be in this case…?)