Instead of a big battery, get Qi charging could and a large charging mat. Now your keyboard is constantly charged when on your desk.
No, you can setup a dongle with bluetooth. In a regular zmk setup with nice!nanos, the two halves communicate wirelessly over bluetooth, a central one that does all the calculating of what key strokes to actually send, and a peripheral that just sends raw keystrokes to the central one. But with this setup, you’d get a 3rd nice nano that you plug into the computer that acts as a central board without any keys that has 2 peripheral boards it communicates with. Because you’ll keep that 3rd MCU plugged into the computer, it won’t need its own battery or even any components, it can just be a bare MCU. You could also make a case for it if you wanted: