DO NOT leave a lithium battery both charging and discharging indefinitely in your daughter's room. The suggestion below of a small UPS device intended for routers and they like is the way to go.

Both of my USBc power banks have passthrough charging and would work for this. That's the feature you need to look for.

However, keeping a battery bank charged to 100% isn't good for longevity, make sure you get a good name brand one so it doesn't catch fire.

Edit to add: the ecoflow & jackery power banks can also charge their batteries while running something. They can do 120v/240v and run the current nose machine, but they're more expensive

They make and sell mini UPS devices. Usually marketed for modems and routers. I use one of those and they have several hours of autonomy for low power consumption devices, and they don't beep uncontrollably like regular PC UPSs.

Most power banks can either provide power, or be charged, not both.

They actually make what you want however, you can get usb UPS modules. They are intended for raspberry pi power, but just present a usb socket. They hot switch between external usb and internal batteries, as required.

Yes Absolutely. Not all of them will charge while plugged in, but I have a couple that do.

Most power banks won't output power and charge at the same time.

There are USB "UPS"s that will do that, though. They're typically used to run Raspberry Pi projects and are quite a bit pricier than regular power banks.

Maybe something like this?

https://www.amazon.com/ENOFLO-Portable-Charging-Compatible-Smartphones/dp/B0BX8R392P

Depends if it is able to power while being charged.

None of my battery banks do that.

I got my son a noise maker that has a built in battery for this same reason. It doesn't seamlessly switch to battery I'm sure there is one that does.

Check out the power bank reviews from the channel "Allthingsoneplace". He does excellent reviews, pretty much exclusively focused in the technical side, testing efficiency, protections, etc. He also tests if it would work as an UPS. Most dont work and turn off after a while.

Yes, battery banks with at least two ports and pass through charging will handle it like this. The Anker bank I have (Anker 737) does this.
For long term use the battery cells may degrade in capacity and max current after a couple years of sitting at 100% charge, but that's probably not too big of a deal for you since the noise machine is probably fairly low power.

As long as the battery bank supplies sufficient power continuously, it would certainly power it! If it doesn't, you'd have to look at a battery bank to power the device but I'd bet your daughter's noise machine is likely going to work fine on anything.

However, skatrek47 brings up a good point: The battery would need to just be powering the machine (unless you had an automatic cutover, but that seems like overkill for this application).

For your use case, I'd say just run it off the battery all the time and charge it once the battery gets around 20%.

There's USB batteries that'll do passthrough charging too, but keeping a battery charged all the time will kill it, so you might as well just use the battery.

If you wanted to go nuts, you could get an APC or something but that would be pretty silly for this.

Would a home battery backup be overkill? If you have frequent outages it might be worth looking at something from EcoFlow ($) or Tesla ($$) to provide power when the grid fails. Other vendors exist besides those two of course.

I don’t think the ones I have would since you have to push the power button to tell it to start charging a plugged in device.