If your Google Nest Floodlight Camera is properly wired and the circuit is turned on at all times, I can't imagine a case where it should be reporting the battery needs charging, unless the camera is defective or the power cord isn't properly seated.
Google Nest does not market the floodlight camera as a battery camera; it's advertised as a wired camera and there is no mention of the battery.
Further, we've had a Google Nest Camera (Battery)--basically the same camera--for 16 months now, plugged in with the optional power cable. We're into our second winter with the camera and have never gotten a battery warning, and our camera is "Live" all the time. Our camera runs off of household current, not off of the battery ("For wired installation setups with a Nest Cam (battery), the wires power the camera directly." - https://support.google.com/googlenest/answer/11830989?hl=en). Your floodlight camera should be the same.
@findyourrare thanks for reaching out here in the community. I'm dropping by to ensure that everything is covered here. How's it going with your Nest Camera? Were you able to read the information and check the link that MplsCustomer has shared above? In case you have an update, you know where to find us.
I appreciate the help, MplsCustomer.