Can the nice!nano (or a different controller) be connected to PC1 via USB for charging and PC2 to bluetooth?

I have a work laptop and a personal desktop. I usually use my PC to listen to music or have youtube on in the background while I work. I currently use a Keychron K2. I leave my K2 plugged into my desktop for charging, but will have it connected via bluetooth to my work laptop. That way, I can just use the BT/off/Cable switch on the side to switch between the two.

I've been doing a lot of research and I want to build the Corne.

Can the nice!nano do that? Or another controller? Could I just power both halves via a wall wart and BT into both machines?