Nah... it was heating up WAY before the gauge told you it was... the gauge reads "normal" from 75 through to 110 or 115 - then it shoots up pretty quick when the engine reaches about 125 or 130 and the gong happens... the gauge is a lying waste of time - although @Young Thrash Driver has my old E39 with the software re-calibrated, so it's a bit more realistic
Also, if the thermostat is dodgy, it may not have opened FAR enough - the flat plate on the back is supposed to block the return port on the rear, and force all the hot coolant into the radiator, and the return into the engine, instead of mixing the two while recirculating coolant in the block as it usually would at "normal" temperatures. The water pump losing drive to the impeller will do it too, as there isn't enough coolant movement.
This is exactly why I changed ours to a mechanical thermostat and re-calibrated the gauge - yours would have been getting hot for a little while before you saw it, then suddenly the gauges and gong go off when it's too late 😞
At least you have a replacement option Maybe you should install a new supercharger while you're in there... the old one probably wasn't working anyway