@Pfb2619
To use the battery doorbell in plugged-in mode, you need to buy the 5-meter or 10-meter optional power cable (https://store.google.com/us/product/nest_cam_weatherproof_cable) and install the camera while it is plugged in, selecting the "plugged-in mode" option rather than the "battery-powered mode" option during the install process. (I can't remember the names of the options; we installed ours 2-1/2 years ago.)
Judging from other posts in this forum, if you purchase the optional power cable AFTER you have already installed your battery camera, you'll need to remove it from the Google Home app, do a factory reset, and reinstall it while it is plugged in, since it is only possible to select the "plugged-in mode" during the install process.
We've had a battery camera since Sept. 2021, plugged in, and it's been "Live" and online ever since.
