After wasting time checking lots of unhelpful advice, I found a solution thanks to Ionuț-Alexandru Popa at binaryfork_. Here's a summary:
- Pair the device (if not already paired)
- Go to Settings > Bluetooth and devices > View more devices
- Scroll down to 'More devices and printer settings' (near the bottom of the window): (this is the crucial step that most of the unhelpful advice misses)
- Right-click the device and choose 'Properties', then the 'Bluetooth' tab
- Edit the name and click OK
- Disconnect the device then reconnect it (this is important to prevent the device name reverting to the default)