Please keep us posted. Did you plan to have a dealer check this? The vehicle is still under warrantee.
Just a guess - but any chance that the previous owner installed something electronic (like a radar detector?). Or - does you cell phone beep when going in/out of range? Reaching here, but sounds like the usual items have been ruled out.
Also, never heard of the clock beep on a BMW before (I've owned 5 BMW's)
It's been a feature in the last 4 BMW's that I have owned, since 1987. It is called the "memo" function.