Is there ASIO driver support available?

Apparently, ASIO allows audio software to gain direct access to the audio device by bypassing the normal audio pipeline.
If I am understanding things correctly, your wish is to somehow introduce additional processing by FxSound into that pipeline.
I don’t know much about ASIO, but I’m not sure if that is at all possible in the first place, since it directly contradicts ASIO’s purpose.
It might be worth browsing ASIO’s own forum or support pages to find that out.

This is a summary of FxSound’s normal audio pipeline,

When FxSound is installed as the only sound boosting program active during audio playback, the audio processing pipeline in Windows is:
Audio application [source] → Windows audio API’s → FxSound audio driver → FxSound application audio processing → Windows audio API’s → Audio hardware driver → Speakers.
But if other audio processing applications are installed, the pipeline becomes more complicated, and isolating the issues becomes difficult.

A first and obvious guess would be, that ASIO interferes with the normal pipeline.

A second fact is that the FxSound Speakers device needs to be set as the Default Device at all times in order to be able to process audio.
So if enabling ASIO turns ASIO into the Default Device, then there will also be an incompatibility on that front.

I also have a few additional questions.

Could you give a description of what happens when you enable ASIO while FxSound is running, or what type of error message you get?

If I am understanding you correctly, you’d like to create music with FxSound enabled.
Now, I was wondering: when you are creating a piece of music, and you have FxSound activated, does the boosted signal also show through in the rendered result? Or only inside the production software?
As far as I know, having FxSound active would only make sense in the first case, since in the second case, it would only distort the signal.

Perhaps it’s also worth mentioning that after a user asked a question about VoiceMeeter software, I found out that it is possible to apply FxSound processing to any track by re-recording it with VoiceMeeter (preferably on the highest desired output settings, of course).
So, that’s something you could try if it turns out that creating music with both ASIO and FxSound enabled is indeed impossible.