Sounds like a bad compressor pulley bearing to me. I'll let someone else chime in on why you're getting cooling on only one side... Could be a few different things.
BMW's solution is complete replacement of the compressor; they do not sell the pulley separately.
You'll need to rule some things out before you can be sure it's just your pulley that's shot. But if you do ultimately arrive at that conclusion, then check out the DIY below that outlines replacing it with an aftermarket part. It's not a simple job though, and it seems like you might have other issues. Lots to check first (AC charge, as well as some values in INPA if you have access).
Pulley replacement DIY here.
I'm sure others will have more to add, but that noise does nottt sound healthy.
You're already living without AC, so if replacing the compressor is not an option for either financial or other reasons, you might want to install a shorter belt as a stop-gap measure to bypass the AC compressor and avoid pulley failure. It happens.
See
330istocknot 's
AC Compressor Bypass DIY here.
Thank you for posting the video too. You'd be amazed at how many people post asking, "what's this noise!?" with nothing more than a crappy written explanation of what they're hearing, and no video or sound.
