I believe there has been many cases of Microsoft Designer Bluetooth mouse users having issues, so as a minimum this product appears to be very fussy.
However, one thing I have seen that does work in some cases is to make sure it's the first Bluetooth device paired to your computer.
In a nutshell you need to reset both the macOS Bluetooth Device List and Bluetooth Controller. However, this will reset all connected Bluetooth devices, including a keyboard and mouse.
In other words, this will remove all existing pairing information for Bluetooth devices, so each will need to be reconnected.
I have some steps you can follow, but I strongly suggest you read all of the steps first before commencing (or perhaps print out the instructions / view them on another device).
Before starting, ensure you have the Bluetooth icon showing in the menu bar. If it is not there, open System Preferences > Bluetooth and enable the option Show Bluetooth in Menu Bar.
Also, since you're trying to resolve an issue with a Bluetooth mouse, you will need to use another mouse to follow these steps:
- Hold down both the Shift and Option keys and at the same time click on the Bluetooth icon in the menu bar. Once the menu is showing, release the keys.
- From the Bluetooth menu, choose Debug > Remove All Devices
- Hold down both the Shift and Option keys again and at the same time click on the Bluetooth icon in the menu bar.
- From the Bluetooth menu, choose Debug > Reset the Bluetooth Module
- Shutdown your Mac
- Ensure your Microsoft Designer Bluetooth mouse is fully powered off
- Restart your Mac
- On your Mac, click on the Bluetooth icon in the menu bar
- Select the Set Up Bluetooth Device option
- On the mouse, press the power button down and keep it down for at least 5 secs. The blue light should start blinking
- Once the Setup Assistant finds the mouse, select it from the list and click on “Continue”.
- Now pair your mouse
- If you have any other Bluetooth devices, you can go ahead and pair them now (or you can do it later). However, the point is to pair the Microsoft Designer Bluetooth mouse first before any other Bluetooth devices!
Let me know how you go.