I didn't know there was a "third class" of HD receivers. Cable cards aside, I thought the choice was either a Mini with its limited tuning capabilities or a full-blown HD receiver (with optional DVR service).
I also thought "client" assumed a host DVR and because of its limited capabilities, you'd use the Mini to watch recorded stuff from the host. When OP wrote "smaller," I thought it synonymous with mini. I'm sure I'm wrong about the Mini and DVR functionality because I've never really thought about it.
You can't reboot a Mini Box?
Granted I can't view all HD receivers on the Cox inventory...("Sorry, this article is not available to you based on your location..." and I don't feel like quasi brute-forcing zips for a peek)...but most look full-blown. Perhaps "clients" aren't listed on the inventory or the distinction between clients and receivers are under the Guide column: Cox TV or Contour.
Your theory would work if the battery was powering all devices. However, if the UPS is in standby mode and has the sufficient number of outlets, the AC outlet juicing the UPS should suffice...unless this UPS is some type of line-interactive model with a faulty inverter. Your theory also assumes all devices within the cabinet are fully powered vs. in either a standby mode or just turned off. Aren't clients and some hosts always powered? Meaning, if the OP turned everything off and went to sleep, wouldn't there finally be sufficient wattage for the client and host to negotiate?
temporarily removing the added device(s) from the old UPS
Why is a cable-box even connected to a battery backup? If a neighborhood loses power, I'd assume the provider node(s) within the neighborhood would also lose power. I highly doubt Cox has independent power sources for its nodes or would even invest in a multi-neighborhood interconnection system...let alone onsite UPS systems for each cigarette-slim pedestal picketing the neighborhood. Aren't these things connected to the closest transformer?
Even if their plant were always powered, Cox would be hawking $30 batteries for all its equipment (cable-boxes, DOCSIS, Pano, Pods, etc) as they do for their telephone modems. I'd understand connecting a media server with mechanical hard-drives to a UPS so you could properly shut it down...but a cable-box? What are you trying to prevent the cable-box from doing during an outage...rebooting? Let it reboot, crash and just swap it. Don't cost nothin'.
I don't know. This is a weird post...although "ghosted" was funny. OP must have crawled back into the UPS-powered bunker.