Many years ago a defunct electronics hobby magazine published an
article I wrote about a simple on-off IR controller. Right after
the detector module I included a simple integrator consisting
of a resistor and capacitor. This created a short time lag which
blocked the transmission of spurious signals. If this new air
purifier didn't have 5 year warranty I would open the darn thing
up and insert an integrator with a nice long delay.

Each press of a switch on the TV remote is very short. The pulse
would have no effect on the air purifier's detector if the
integrators time interval were two or three times longer than a
typical press on one of those membrane switches.

You sound a handy sort of chap - it sounds like you need simple
IR repeater, photodetector followed by an IR LED with an RC
filter to get rid of the short pulses from the TV remote. The
IR LED will need mounting in front of the purifiers detector
with black tape (or whatever) so that the purifier can't see
the TV remote.

The repeater may involve a little more complexity but no more
than a nice low power OPAMP.

Useful?
---Paul