The Eaton unit you listed shows the battery capacity as 2200mAh, which may not be enough time for your controller during an extended outage. The website does show a run time graph which may tell you if it will work in your situation.
You might consider a DIY approach using a lead acid 12V battery and a power supply that can float the battery at 13.2V to 13.3V and occasionally raise the voltage to 14.5V for a few hours every couple of weeks. A number of marine rated battery chargers use this method to keep boat batteries from degrading when not in use. An automotive battery that doesn’t have quite enough capacity for winter starting could be repurposed in your situation giving days of backup.