I have the hardware, but I use SDR# as HDSDR interface was not to my likely a few years back when I tried it. The latest ver2.76 might be much better now.
It does look like on the HDSDR website airspy is supported using an EXTIO.DLL file. Looks like a standard version of EXTIO and enhanced version is available. As I recall place that DLL in the same location as the HDSDR executable.
Another resource is
rtl-sdr.com - RTL-SDR (RTL2832U) and software defined radio news and projects. Also featuring Airspy, HackRF, FCD, SDRplay and more. if you have not already found it.
There are a few tutorials generally specific to lower level hardware. From those tutorials I was able to pick and choose what might work, but in the end I found it to be trial and error.
Try it with just the Airspy to see if that works OK with HDSDR. Add the Spyverter next. I forget if it has a software selectable bypass mode and I don't recall the Spyverter's offset frequency.
Sorry, no plug and play instruction, but maybe I have given you a direction to pursue that might help.
Good Luck!
Mike