You cannot use a USB battery! A USB battery is 5V and the basebox requires between 12 and 24V. So it won't work.
However, you can use that exotic device that you showed the Amazon link for.
It can output the correct voltage. But probably it will have the problem I mentioned before: you cannot wire everything
up and wait for the power to fail and the battery to take over automatically. You will have to have the basebox on its
original powesupply and the battery on a charger until the power fails, then you go to the basebox, remove the plug
of the original supply and connect the battery, and it works on the battery until it is empty. When the power comes
back you re-wire everything as original.
This is not convenient and it would not be required with the mUPS, where you install everything and it will automatically
switch over when power fails, decide when to charge the battery, etc.
Also, you probably want to know how long it will operate on the battery, and there is really no way of telling that.
Why? Because the manufacturers of those batteries are known to be outright LIARS. The specs they mention on
the website are inflated because they know the customers go for the highest figure and will not have the knowledge
to later check if the product they bought complies to those specs.
Example on your link: in the header it says "Weiyi 40000mAh Ultra Slim Portable Power Bank".
When you scroll down, in the details it says: "Specification Capacity:33600mah(124.32Wh)"
See how they already have rounded up the 33600 to 40000?
But, the Wh spec indicates that the 33600 mAh spec is at 3.7V, the voltage of the actual battery in the brick.
When you select an output voltage of 12V, the Wh number remains the same (when the converter is 100% effective)
so the mAh number at 12V is only 124.32Wh/12V = 10.36Ah = 10360mAh.
That is quite a lot less than 40000. And I guarantee you that when you buy the unit and test it, it will be even less.
How much less, well it depends. Could be 4000 or so when it is a really crappy manufacturer, or 8000 when he is honest.
But he knows that 4000 or 8000 does not sell, and 40000 does, so that is what goes on the label.