Hi,
Bluetooth devices are different from wireless devices. You cannot connect the Wireless Mobility Mouse using the built-in Bluetooth feature. You would have to use a transceiver to establish connection.
Transceivers are not sold separately. I’d suggest you to contact Microsoft using the phone support option and explain the scenario.
Refer these articles:
How to replace Microsoft software or hardware, order service packs, and replace product manuals
Support for Microsoft mouse, keyboards, LifeChat, LifeCam & more