Here's a moronic OCD setup I want to have, is there anything like the receiver/transmitter paired USB dongles I'm imagining in pink in the middle section here?

Alternatively I hear most tri mode keyboards can connect to two PC's at once via different input methods and switch between them via switch or keybind, is that true? Or would I need to look at specific models? Do any tri-mode keyboards have removable batteries or take normal AA batteries if I'm worried about lithium battery swelling?