Replace or recharge the mouse battery even if it "seems fine"
Device Manager → Bluetooth → your adapter → Properties → Power Management tab → uncheck "Allow the computer to turn off this device to save power."
Do the same under Device Manager → Mice → your mouse entry, if it has a Power Management tab.
Remove the device entirely in Settings → Bluetooth & devices, restart the Surface, then re-pair.
Update Bluetooth drivers via Windows Update → Optional updates, and check if your mouse manufacturer has firmware/software updates (Logitech Options+, etc.).