As long as you aren't killing the battery by constantly viewing its livestream (either via app or the TV), then I think I would try connecting the suspect camera to the same base station as the working camera. Each base station can support 4 OC cameras.
I will add that in the first week I had the OC, the battery drained pretty quickly, but subsequent charges have seemed to have stabilized it. I have also noticed that the mobile app does not always accurately report the battery strength. I have had it plugged in charging and the app kept saying it was around 60% until I removed the SmartHome app and reinstalled the app. Immediately it recognized it as 100%.
It does indicate that longer the cooldown, the longer the battery is anticipated to keep the charge. I'd much rather have a dead battery than allowing someone 3 or 4 min to get through the door until it decides to start recording again, though.
No matter what the specs say, "With normal use, the rechargeable battery in your camera will last 3 to 6 months" (https://support.roku.com/article/7869544173719), I am not seeing this at all and will likely just be adding a solar panel to this OC to keep from having to charge the camera every month. (However, it may perform better during some warmer weather as freezing temps and batteries are not usually a good combo).
