OK, not having the custom USB Map configured is an issue that commonly screws with Bluetooth, as the device is powered over a USB connection.
Moving from Monterey to Ventura shouldn't require any changes for the Intel BT device. As the last major change to the BT stack was when we moved from Big Sur to Monterey. Bluetooth under Ventura and Sonoma should be the same as Monterey.
For Monterey and newer these kexts are required.
- IntelBluetoothFirmware.kext
- IntelBTPatcher.kext
- BlueToolFixup.kext
Use/place them in the order above in your config.plist.
If you have added these kexts to your OC setup, then it is probably the lack of a USB Map that is preventing the BT device from working.
From my reading it appears that the HP EliteDesk 800 G3 contains the following USB ports.
Front
1 x Type-C
2 x USB 3
Rear
4 x USB 3
Internal
1 x USB 2 Internal port served from the M.2 WiFi header.
Assuming your system has the same USB port configuration this should be a simple USBPorts.kext to create, as the total number of USB ports will be 15 or 17, depending on the front Type-C port.
If you don't currently use or plan to use the Type-C port it can of course be omitted from the kext. But I would try to map it just to be on the safe side.
- There are no physical USB2 ports, so none of the ports would be set with connector type USB2 (0)
- There would be a maximum of 12 x ports set with connector type USB3 (3) (2 x each from the 6 physical USB3 ports)
- There would be 1 x port set with connector type Internal (255) (Bluetooth)
- There would be either 2 x ports set with connector type Type-c+sw (9), or 4 x ports set as Type-c (10).
How you tell the difference between the two Type-C variants is as follows:
- Insert a Type-C USB pen/drive in to the front port.
- A single USB port will be highlighted in Hackintool or IORegistry Explorer
- Remove the Type-C device, flip the device 180° and reinsert it in the same port.
- If the same port is highlighted, then this is a Type-c+sw port (9)
- If a different port is highlighted, then this is a Type-c port (10)
Type-c+sw (9) - will show two ports being available, 1 x Physical Type-C and 1 x virtual USB2 port.
Type-c (10) - will show four ports being available, 2 x Type-C and 2 x virtual USB2 ports.
I would strongly recommend you try to create a USBPorts.kext for your system.
You would need to enable the Kernel > Quirks > XhciPortLimit entry in your config.plist and use USBInjectAll.kext (doesn't need to be a new version as this is a Skylake system) so all the ports on the system are available for discovery.