SIP works, but excepting a native app from Doze doesn’t. For example, if an app has its own push notification system, even if it is whitelisted, it will be terminated during doze.
EDIT: Yup, https://romhut.com/versions/cm-13-0-20160508-v8-0 indicates it's been in CM for a while, which is probably why I haven't experienced it. Dianne Hackborn fixed this as far back as November 2015, it might have made it into 6.0.1 on some devices too, though I could be misreading.