tl:dr What is a safe way to connect a simple plug-in UPC battery back-up so that it can power a single receptacle on another side of a wall?
We have an older farm house that has been expanded a few times. I have a second wifi mesh router that I want to put on a shelf in our kitchen to improve the wifi. The shelf is about eye level and I will put a power receptacle on the shelf along with a CAT6 connection. I can fish the wires easily enough for the new outlet by going down the wall and connecting to an existing outlet that is on the opposite side of the wall (and hidden in a closet strangely enough).
Anyway, what I really want to do though is have this wifi mesh router on the shelf connected to a UPC battery but there isn't enough space on the shelf (and it would be an eye sore anyway, the little wifi puck is tiny).
So how can I power just that outlet from a cheap UPS?
Idea one: cut off the female end of an extension cord and hard wire it to the new outlet on the kitchen shelf and fish the male end down and out the bottom of the wall (hidden in the closet) and plug into the UPC. (Downsides: male plug stickup out of hole in the wall!)
Idea two: put a second outlet in the closet next to the hot outlet but only wire this new outlet directly to the shelf outlet (so the outlet is not actually hot - just two outlets connected together). Then use a male-to-male power plug and connect the UPC battery backup into the new outlet so it can power the plug above it on the shelf. Like: reversing the flow of electricity.
The second idea seems a little tidier and less gaping holes but also seems stranger so I wonder if there is something I am not thinking of.
Appreciate anyone's help!
Edit: The wifi router I am using is a Google Nest Router (not "point"). The back of the router indicates
14V ⎓ 1.1A.