You see that crescent shaped lever with all the teeth on it? That is what the motor acts on to raise/lower the window. You might try to either pop it off the motor or use one hand to push the window down while pushing up on the lever to lower the window. If you can get the window half down you can disconnect the regulator and pull the window all the way up and jam some wooden shims in there to hold it in place. You do not need to remove the glass on a convertible.
Let's talk about the 3 wire connector for a second. We know that we have voltage to the motor when the switch is activated, the next thing would be to figure out why the BECM is cutting the power right away. Those 3 wires are for a potentiometer, which is how the BECM knows what position the window is in. Basically, a potentiometer is a resistor with a reference voltage (usually 5v), a ground, and a signal. The BECM sends the reference voltage through the resistor and out the ground. The signal gets a varying voltage depending on where the window is. It's possible that if there is an open in the ground or signal the BECM might think the window is already in it's lowest position and cuts power to the motor. I can't tell from my schematic which one is the 5v reference and which is the signal, but it's one of the black wires. The brown wire is your ground. So with the key ON try to get a voltage reading on the black wires, you should get 5-9v on one of them (I'm not sure what the specs are). If you're getting voltage the next step is to make sure we have a ground, so we want to Ohm the brown wire. It should be nominal resistance, like .25 Ohms. The other black wire is a little tricky because I don't know where it goes inside the BECM, but if the key is on and it reads OL or ridiculously high it's a problem.
Something else that I came across looking at BMW DIS was a document talking about how the driver door can get stuck if you disconnect the battery. When did this problem start? There's something to do with the convertible top that allows the window to get voltage through a circuit breaker (I think it was labeled F56 but I couldn't locate it) before the BECM comes online. Anyway, the fix was to trip circuit breaker F56 (I think). If you can locate the circuit breaker in your Bentley manual it might be a quick fix.