All of this is being insanely overcomplicated.
Throwing more complexity at a simple problem might be "fun" from a nerd's POV, and, TBH, building this USB device sounds fun. But entirely unneeded while introducing more points of failure.
A simple solution to your problem:
1. Get a monitor with a built-in USB hub (nearly all of them?). Consider getting a USB-C monitor to reduce the number of cables to 1.
2. Don't use Bluetooth (for a keyboard, for multiple reasons, like needing the keyboard available in early boot). Get a keyboard/mouse with an external USB dongle like Logitech's Unify or Bolt, Corsair's SLIPSTREAM, or any of the other billion options that exist.
3. Plug keyboard/mouse into monitor, plug random computers into monitor. Bam. Unified mouse and keyboard without any pairing.
Throwing more complexity at a simple problem might be "fun" from a nerd's POV, and, TBH, building this USB device sounds fun. But entirely unneeded while introducing more points of failure.
A simple solution to your problem:
1. Get a monitor with a built-in USB hub (nearly all of them?). Consider getting a USB-C monitor to reduce the number of cables to 1.
2. Don't use Bluetooth (for a keyboard, for multiple reasons, like needing the keyboard available in early boot). Get a keyboard/mouse with an external USB dongle like Logitech's Unify or Bolt, Corsair's SLIPSTREAM, or any of the other billion options that exist.
3. Plug keyboard/mouse into monitor, plug random computers into monitor. Bam. Unified mouse and keyboard without any pairing.