As you can expect: "nothing is new under the sun…"
A different variant has been used locally…
First, a note on your proposal, the secondary part: Your system would upset the motor’s thermostat’s ability to stabilizise working temperature, if this system would be in series with the motor’s main circulation loop.
Even in parallel, control is hard to achieve.
The circulation direction in your secondary (motor bound) loop is opposite to the motor’s normal circulation route.
It feeds out warm water.
Additionally the temperature in the motors cooling system will not (usually) go low enough to cool (indirectly) the “washing water” in your system’s primary loop.
How do you handle the primary loop in the winter???
The local system was an independent (in the back of the car) self-contained system, only using electricity for a circulation pump (trailer heat) and delivering cool gas to the car motor.
No primary, direct washing part. Only a cyclone and an indirect cooling “battery”, liquid cooled with a car-type radiator and the circulation pump. System filled with 50/50 glychole/water.
After cooling and settling, reheating in the cyclone’s mantel, then to a warmed paperfilter. Then cooling again on it’s way to the motor, but that does not affect the filtering anymore!
You did not mention how to dehumidify,
nor did I… almost not at all.