TPMS or a low pressure warning system? DOT requirement on SUVs was only for the latter, some manufacturers monitor pressure, but maybe Go-NoGo, others use the antiskid sesnsing equipment to identify a low tire based on rotation speed differences. I have a 2013 Ford that does this one way, and a 2012 Honda that does it the other,
Either way it is handled through the car's compute, which has to be running to process the inputs into warnings at the instrument panel or information center display. Engine may or may not need to be running, it is hard to say without knowing just how things are integrated for a particular model.
If the car is using pressure sensors, and those use radio to report to the engine control or stability systems, it is still unlikely that a third party TPMS will pick up the signals and understand them properly.
