This admittedly isn't software. I'm leaning on the shoulders of giants in the professional teleconferencing space for this. I'm using a beastly old Biamp Nexia VC audio processor for this. The nice thing is that these are very end-of-life products that don't require a license to operate, so you can pick one up on eBay for $50-100.

The Nexia provides Acoustic Echo Cancelling, which is fundamentally the same stuff that VoIP apps like Zoom, Teams, Discord, etc. use to detect feedback and squelch it, except instead of ducking my microphone, it does waveform cancellation to strip it out, to pretty great effect. To take advantage of this, I send the audio output from my computers (I have my desktop or my laptop, which are attached to the same USB audio interface, and a Mac Mini that I use for secondary tasks, media watching, and a CI/CD worker target) as well as a contact microphone attached to the wall adjoining a bathroom. Audio from those sources are then removed from my microphone feed before it goes back into my audio interface for whichever primary computer I'm using as my teleconferencing device.