So it looks like you might be able to use the built in transceiver but you might not get the extra functionality that that mouse has.
Use other Bluetooth transceivers
You may be able to use another Bluetooth transceiver, such as one built into a notebook computer, to operate your Microsoft wireless keyboard and mouse.
However, your other Bluetooth transceiver must use the Microsoft Windows Bluetooth software to provide all the features your keyboard and mouse has to offer.
Different transceivers on the market may use one of several different versions of Bluetooth software, but only the Microsoft Windows Bluetooth software provides full support for your keyboard and mouse. To find out what Bluetooth software your current transceiver is using, see the documentation included with it.
Depending on your other Bluetooth transceiver features, you may be able to change the software it uses to the Microsoft Windows Bluetooth software. This more complex scenario is not supported by Microsoft, and you may have to reconnect your existing Bluetooth devices if you are successful.
Using a different transceiver for your Microsoft Bluetooth keyboard or mouse is not supported by Microsoft. For more details, see the documentation included with your other transceiver.