The bleed procedure as published by BMW is for a dry fill from factory when they need to prime the secondary water pump etc.

For routine bleeding, such as after cooling overhaul (WP replacement etc.), I just do simple things:

1. Engine Cold and Off.

2. Open bleeder screw near the tstat (remove it but don't lose it).

3. Fill the reservoir with the 50-50 mixture until it is 1" above KALT or roughly about 2" from the rim. Now, be patient! It is a slow process and coolant will flow out of the bleeder hole.

4. Re-install the bleeder screw (just snug it, don't force it!), then make sure coolant is still about 2" from the reservoir rim.
Replace the cap.

Now go for a test drive in the neighborhood with heat on etc., drive until the engine is hot.
Then go back home, park the SUV.
Wait 1-2 hours, then check coolant level, it should settle down to MAX area.

I have done this trick many times (overfill just a tiny above KALT when COLD) and it works. Very very easy.

If you overfill the pressure will build to either lift the 2bar (29psi) cap or burst / break something. An older system will typically have a part failure before the cap lifts. Many install a 1.2bar (17psi) cap. Better to slightly under fill than overfill.
A 50/50 glycol mix at 1.2bar has a boiling point of 270F. A 50/50 glycol mix at 2 bar is about 295F ( couldn’t find the exact number so used the estimate of 3F/psi from 280F@24psi)

Where would one buy a 1.2bar cap that fits our reservoir?

I’ve been using an e30 1.4 bar cap for 5 years now. I feel way safer with it. The only retarded car in the world with a hose / expansion tank / radiator / valley pan exploding 2 bar cap.

Cool, thanks for the pointer! Got it from dealer for only $17.