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I dunno how much power those things draw, but a UPS designed for a desktop computer would be perfect for a situation like that. They come in many different capacities, just get one that matches the size of the equipment.
I wasn't really aware of this until a few years ago, but UPSes aren't really generally designed to keep computers running through power outages. Even today, most still have lead-acid batteries (which have limited energy density and don't deal well with being deeply discharged many times).
They were really designed to solve two problems:
back in the 1980s and 1990s
shut down cleanly, to avoid filesystem corruption, when commonly-used filesystems could become corrupt through not shutting down cleanly. They also often came with functionality to automatically shut down the computer cleanly when the UPS's battery was getting low, if a human had not yet done so.
This means that UPSes tend to have a pretty decent inverter, can put out a lot of power...but generally can't store a whole lot of power. They also are guaranteed to come online quickly; I believe that it's typically in under 20 ms, a blip that a computer power supply can handle.
To grab a random UPS:
https://www.amazon.com/APC-SmartConnect-Interactive-Uninterruptible-SMC1500C/dp/B077Y62GSJ
That runs $529. It has a 900W inverter.
It uses this battery, which is lead-acid 11Ah 12Vdc (so 132 Wh).
To grab a random power station of about the same price:
https://www.amazon.com/BLUETTI-Portable-Station-Generator-Off-grid/dp/B095Y6ZTR1
That's $500, has an 800W inverter, but has 716Wh of battery storage, about 5.4 times what the UPS does. It also has an LiFePo4 battery, which will last a lot longer in terms of cycles and dealing with deep discharge than a lead-acid battery.
I think you are kind of comparing apples and oranges. A decent UPS delivers voltage regulated wall power (smooth spikes and dips, or a brownout) and seamlessly switches over to battery power when wall power dips or drops. It often has a bunch of additional hardware that adds to the cost but wouldn't be useful to someone using a CPAP machine with it. Ex. remote access via Ethernet or at least a connector to signal a PC so that it can cleanly shut down.
I don't believe that power stations like the one you linked are designed to maintain power seamlessly during an outage. I recall it being a battery that you'd switch to if you lost power or to take camping. I don't think it's meant to be run plugged in all the time with devices drawing from it all the time. Admittedly, I have no power stations so maybe I'm not the best person to comment.
I guess you'd have to wake up to switch it over, but honestly just a power bank with a standard outlet would probably be adequate.
https://homegrail.com/how-many-watts-does-a-cpap-use/