The VGP-BMS33 uses Bluetooth 2.0, which is power-sensitive. Replace the AA battery with a fresh alkaline battery. Also, go to Device Manager > Bluetooth > your Bluetooth adapter > Properties > Power Management > Uncheck "Allow the computer to turn off this device."
There is a specific breed of tech enthusiast—or perhaps just someone who is incredibly stubborn about not buying new batteries—who still clings to the peripherals of the late 2000s. Sony Vaio Bluetooth Laser Mouse Vgp-bms33 Driver
if the standard Windows pairing isn't working for your specific OS? Searching for driver for Sony VGP-BMS33 - Microsoft Q&A The VGP-BMS33 uses Bluetooth 2
Sony VAIO VGP-BMS33 is a classic Bluetooth laser mouse designed to complement the VAIO laptop series with its sleek, ultra-modern aesthetics and cable-free convenience The VGP-BMS33 uses Bluetooth 2.0