I believe it will still work, but not 100% sure. When you set it up, you indicate if it will be hardwired or battery, I think it uses that when providing some settings.
I also think it will function similarly to the BCP. I believe the camera runs from Battery and the Power keeps it charged. I am not sure it will run with the battery out. let me check a few things and get back to you on it
EDIT
I appears that if power to the doorbell is lost, it will continue processing and functioning. However, it does seem that the Continuous recording will stop for the period of time it has been off power. When you add the power back, it seems to pick up and start continuous recording again.
But events are recorded to the SD Card, it just wont be continuous