So to confirm:
- starter seems good
- key moves to ignition position 3
- still no signs of a crank
If you have BMW diagnostic software, I think it'll show you if it's meant to be cranking etc. My campervan did this, and the Mercedes software told me it was getting start authorisation from the ECU, so I knew the issue was with the starter, and not the ECU/immobiliser - see if you can find the same thing with the BMW software. (I hit mine with a hammer and it has been fine since.)
Some thoughts that spring to mind that I haven't seen mentioned:
Try jump starting it, just to rule out battery after standing (despite what the voltage shows)
Is it an auto? Does it know it's in park?
Have you checked for 12v at the wiring to the starter motor?
I don't know if there is a starter relay on these, but if there is, try jumping it to see if the starter fires.
Have you tried bypassing the EWS to see if it'll crank? (I'm at work, so you'll need to do more research into this route:
https://www.bimmerforums.com/forum/...s-)-the-EWS-to-activate-the-starter-on-a-swap )