For the Bluetooth 4.0 version, the HK-808 is a standard "generic" USB dongle, and the key to success lies in its underlying chipset. The most probable candidate is a , most likely the CSR8510 model, which is ubiquitous in many affordable Bluetooth 4.0 adapters.
Often, users share CSR4.0/5.0 drivers on forum sites. Search specifically for "CSR 4.0/5.0 macOS driver" as this is the chipset inside the How to Install the HK-808 Bluetooth Adapter on Mac
The kext (kernel extension), which is part of the broader BrcmPatchRAM or OpenIntelWireless ecosystems, can sometimes inject compatibility for Realtek devices. For Hackintosh or OpenCore Legacy Patcher (OCLP) Users:
HK-808 Bluetooth USB Adapter Driver for Mac: A Complete Guide The HK-808 Bluetooth USB Adapter
Given the age of the HK-808 and the increasing challenges of driver support on modern macOS, the most practical advice may be to upgrade.