Any idea if the opposite exists: a dongle that appears to the host as a USB keyboard/mouse, but actually pairs with a bluetooth keyboard/mouse? That way you could use a bluetooth keyboard with dumb KVM switches (or other USB hosts) that don't support bluetooth themselves.
You have too high thoughts of my kvms :) It's a Blackbox KVMoE system which seemingly only understands standard USB keyboard/mouse. I guess it depends on the USB stack in the actual switches. I will have to try with a pre-paired Unifying receiver though, perhaps that works.