Do I need an Oil Cooler Thermostat?

But if you have a thermostat with a heat exchange for coolant/oil then wouldn't losing coolant be like losing your oil radiator and coolant radiator at once?

Well... the engine would definitely overheat quickly.

But it would not be like losing both, because I would not be losing any oil, which I would if I had an oil cooler and lost it.  That would stop my engine dead.

Then I guess it might be even safer with a heat exchanger and only the radiator for coolant than without because the heat exchange is less likely to fail than it is for the oil radiator, which is the opposite of what I thought at first. That's an interesting take. Why doesn't Rotax do that on their own? I bet it would save weight.

How quickly would the engine overheat at typical cruise power if coolant pressure went to 0 and the oil stopped getting cooled as well?

I don't think Rotax mandates any particular set of radiators/coolers.  It will depend a lot on each aircraft's configuration.  In my case, there is no cowling, so I never see either coolant or oil ever getting even close to maximum.  In my configuration, a separate oil cooler is totally unnecessary.  As it is, I'm blocking off airflow over part of my radiator in all but the warmest weather.

If I totally lost coolant, I imagine things would overheat very quickly, and I would be looking to shut it down as soon as a safe landing looked assured.  But, for example, if it were to happen on takeoff I'd try to keep it running, even if it ended up costing me the engine, if I thought it would help me save myself and the rest of the aircraft.