Personally I use a turkey baster or something. I avoid pulling the radiator drain as it will let out quite a bit of water and is hard to control the flow/ get it back in. I probably go for the expansion tank bottom hose if not turkey baster. Do it when the engine is OFF and fully COLD, the same as testing the coolant level.
For the leak check under the expansion tank, where the sensor is. If you had a replacement expansion tank from Behr fitted, 8 months is bang on schedule for it to fail. You'll need to replace it with a genuine BMW one. Also check the metal/plastic joins on the radiator, these are a failure point even on new radiators. Otherwise it could be your valley pan gaskets, normally leak slowly and the water runs down the transmission area.
An e39 is like a beautiful wife, needs constant attention, lots of money, lets you down at the worse times but you’d never be without it.