These rigs are a pain. So I can't be sure if this can be fixed.

But let's try a thing. Most TVs will not pass any HD audio over to Optical so I'd try an Optical link to the soundbar and maybe that will mute it when playing from the HTK.

Or get an HDMI audio extractor (so many so I'll link to one with no intent of guarantee that's the right one) to put in line with the HTK to HDTV to drop the audio out. Example:
https://www.amazon.com/ViewHD-Extractor-Optical-Toslink-Converter/dp/B00KBHX072

Sorry about the fact I can't be sure what is the right way to work this one. I could write about receivers but I understand you want to try to fix what you have.