I can imagine this has been an inconvenience, and I would love to look into this further for you. I do however need some additional information to best assist you. There is no way for you to test the internal battery for the Nest Doorbell (wired), formerly known as Nest Hello. It is a wired setup, so as long as it is set up, the power provided charges the internal battery.
I just wanted to check in to see if you had seen my last response. Just as a reminder, there is no way to replace the internal battery in the Nest Doorbell (wired). If it has failed, then the whole device has failed, most likely. Would you please explain more about what symptoms you're seeing?