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Very interesting. So theoretically you could put a real LSD in, and just run it on the last setting and be fine.
The ELSD is not needed in the other modes. When the car is first started it in in "DSC mode". This is the most intrusive mode and keeps all four tires from "slipping" and uses the yaw sensors to keep the front and rear "inline" i.e. keeps the car from slipping sideways. When you press the button for 1 second it turns off the DSC, but leaves the DTC on. This basically shuts off the yaw sensors, but still keeps all four tires from slipping. So the car can "slide sideways" but keeps forward traction. The last mode invoked by pressing the button for 3 seconds shuts off the DTC (and DSC) but leaves the ELSD on. This allows the front tires to rotate at a different speed then the rear tires BUT tries to keep the two front tires rotating at the same speed. So as you can see in the first 2 modes you don't need the ELSD anyway (or you can say that it is on always).
So a Mechanical LSD would not help unless you can turn off the ELSD in the last mode