Even though you technically
could do this with a wireless USB hub, you're going to find that the WhiteFox isn't designed for power saving. Batteries are going to get nuked and the whole setup is going to be clunky. You'd need a reasonably bulky battery pack and even if you did your own soldering and 3D printing, a fully-custom solution would likely need to add two or three 18650 batteries to the keyboard, ruining it's compact size and shape.
Wireless keyboards are designed to be power-efficient. Wired ones are not. Everything from the choice of integrated circuits to the internal voltages is optimised for battery life in a wireless keyboard and when you run something off a battery that was originally designed to have access to a limitless 5V, 250mA supply, it will use it because that's the simpler/cheaper/easier to build for.
A wired keyboard doesn't even have the power management hardware or firmware it needs; There's no concept of power-saving so it will use order(s) of magnitude more power than a purpose-designed wireless keyboard. Even if you didn't
use the backlighting, I'd be surprised if the wireless USB bridge and keyboard itself didn't drain a 5000mAH battery pack in a couple of days of use - oh, and if you want backlighting I'd be amazed if it lasted more than a few hours. The keyboard may not even be fully to blame, either - I have used several wireless USB bridges and I have never seen one that wasn't designed for a mains-powered device. Maybe power-saving, made-for-battery wireless USB bridge solutions exist, but their market niche must be vanishingly small which certainly means it'll be expensive and hard to obtain
It's possible in a "proof of concept" sort of way but it would be hopelessly impractical and genuinely expensive to attempt.