Possible to use two HCI controllers concurrently?
After some back and forth with Apple Support I've learnt that yes, macOS can use two Bluetooth HCI controllers at the same time, but the same driver cannot be used for multiple devices at the same time. As both my internal Bluetooth module and the USB dongle I purchased use a Broadcom chipset, macOS uses the same driver and that's why I cannot use both at the same time.

The solution is to use a USB dongle which does not contain a chipset made by the same manufacturer as the internal Bluetooth module. Broadcom seem to make the majority of chipsets used in most Bluetooth USB dongles - however I've found the 'Orico BTA-403-BK' which uses a CSR8510 chipset, which is manufactured by Qualcomm, who acquired a company called Atheros in 2011, whose Bluetooth drivers are natively supported by macOS. I'll report back after the Orico arrives with my final result.