I have the exact same problem regarding the doorbell press notifications and sharing the device via the Hik-Connect app. When the doorbell button is pressed, I get a video call request on my phone, but my wife does not get the same on hers. She does get the motion detection alerts though. This renders the Hik-Connect app largely useless for my needs, so I'll look into monitoring the doorbell button press externally and generating a push notification including a still image from the DS-KB6003, based on this:
DoorBell + Moteino = Awesome | LowPowerLab. The DS-KB6003 can then be isolated from the internet entirely.