None of Anker's batteries charge and provide power at the same time so they cannot be used as cheap UPSs.
I think you'd be better off looking at some of the DIY electronic battery things - eg WiMos battery shield or adafruit powerboost (https://www.adafruit.com/product/2465) and a LiPo battery, but they will only provide ~1 Amp.
I think I have an APC Back-UPS 425VA (or 600VA) that takes care of that for a sprinkle of RPis, a firewall, an access point and the ONT outside. I like to keep my connectivity during the very brief power outages (underground power).
I've been meaning to try using an Anker portable battery as an extended-run UPS, but haven't tried it, and I don't know if it'll keep the USB port powered on continuously.