Battery, cables, starter. The CPS has no bearing on an engine not cranking. This is most likely the battery. Have it tested as suggested. I agree that it is not uncommon for cold weather to kill a battery.
Current ride... '13 GTI Wolfsburg
BMW's from my past...
'92 535i/5 '89 325ix/5
'98 750iL '86 325(e)/5
'95 740iL '84 318i/5