Older BMW I’d want to be in the 40-weight range for viscosity.
If you want to use up the oil you have I’d blend it so you end up in the high-30/low 40 viscosity range.
If you are ‘starting fresh’, can’t go wrong with M1 0W-40
HTHS of that product is under 3.5. (Technically it 2.9 - 3.5 but usually they fall around the 3.2 range)
BMW spec'd an HTHS of 3.5 or higher (i.e. A3/B4, LL01).
Now that doesn't mean people haven't run A5/B5. Back when I had an E46 I would occasionally read a poster who said they had no problems running vanilla M1 5w30 but whatever. It's not like they'd pop back in and say "well my bearings failed, I guess y'all were right all along".
I'm wondering when it's appropriate to switch from xW-30 to xW-40? Is there a reason to make that switch? I live in New England, and we do get many days below freezing, and occasionally into the single digits Fahrenheit.
(BTW - The BMW is now mine. It's sitting in my driveway while I jump thru the hoops to register and inspect it)
If you are not commuting at 165km/h + and you don't have any rod or main bearing clearances outside the tol. window, you should be good for 5-6k mi OCI. Are these engines built so terribly that they would blow up where Ford and GM high pressure turbos live with 30 and 20 grade ILSAC lubricants?
Just listen for odd and excessive noises when fully warmed ... If it gets too rackety, drain and install a A3,B4 30 grade. I would also not recommend "autocrossing" events.
Now about those electric water pumps ...
-Ken, Your neighbor across the border to the North.