Knowing that you car was a 124, I purposely included the qualification that my direct experience was with a 126 chassis, and because the 124 chassis is not a virgin design you will be happy to know that many proven design elements of earlier chassis’s where incorporated into it.
I am fortunate to have a 124 ETM and I have taken the opportunity to investigate and you can be assured that there does not exist an electrical nor electronic control mechanism regulating the washer fluid temp. There is a washer fluid level switch attached to a grommeted sending unit in the washer fluid tank. And there are two electric pumps, one for the windshield nozzles and the other for the headlight wiper nozzles of course.
From a 1989 300 SE 126
Heating of tank by coolant similar to the windshield heater, the heat exchanger (tubular coil) in the washing agent tank of the windshield washer system is supplied with warm coolant from the bypass flow to the heating circuit. A thermo-valve with a wax pill element located in the washing agent tank ensures uniform temperature of the windshield washing agent of +20C to +30C”
Do you think that in the same model year MB might have used the same system and components in two different but related chassis? Your over-heating would appear to be possibly related to a failure of this in-tank mechanical thermovalve.
The thermo-valve that I spoke of is a mechanical valve that is designed and functions to open at +15C and is completely open by +4C outside ambient temperature and below allowing warmed coolant to the windshield glass heater. Again it would not surprise me if the warmed coolant that flows through the heat exchanger/thermo-valve inside the washer fluid tank is provided from the ambient air temp thermo-valve that controls flow to the windshield glass heater! Since an ambient thermo controller and flow is already engineered and installed why wouldn’t those wild a crazy Germans expand and utilize it? It would be much less likely for such engineers to design and install a much more failure prone electric/electronic system when a hardy mechanical system would suffice, don’t you think?
The heat exchanger/valve in the washer tank receives its flow in parallel to other devices and functions I would bet rather than a series arrangement, so the mimimal partial obstruction of its flow would most likely be without adverse consequence. You could always simply pinch it off temporarily to experiment. At low engine speed the electric auxiliary coolant pump directs coolant flow through the monovalve/heatercore/bypass flow circuit and stopping the small flow through the washer tank heat exchanger would have negligible effect.
Perhaps some other 89 300E owner will be able to comment whether or not the washer fluid is always heated rather than only those times when the outside ambient temps would make it useful?
The cabin heat problems I metioned where from other previous threads you had ether posted or replied to. And yes, the relatively small investment in the documentation for your car would help avoid circumstances such as these when no one can provide the exact and specific answer for your particular problem of the moment. Good Luck!