There is a way of doing this.
My solution was to use Dante Via.
I use various headsets, including Logitech PRO USB-based streaming headphones, and wireless ones, for teaching at university, and private online courses as well.
To get audio output to ANY headsets, and to get microphone input from them, I drag the respective headset source (mic) to the streaming app receiver in Dante Via, and the streaming app source to the headset receiver (i.e. headphones).
You can also route practically any number of other audio sources to the streaming app receiver and/or the headset receiver.
So you can get TotalMix FX channels into your streaming headphones like that, even if they are USB-interface based or wireless. This includes great gadgets like the professional Logitech streaming headphones, but also Bluetooth models from Plantronics or others. Even Chinese cheapos work well like this.
For redundancy (battery precaution measures for wireless etc.), I always route at least two headsets like that in tandem. If one should fail, I can always pick up the other one. Though that has turned out to be a bit of overkill. So far, I've never needed to do this yet, fingers crossed.
Latency is no problem whatsoever. Streaming latency is relatively high even without Dante Via anyway, and Dante Via synchronizes the listener's experience and input/output with the streaming input/output.
The above setup works perfectly on Mac and Windows. I use a Windows machine as main videoconferencing/streaming machine, and a MacBook Pro as a redundancy backup -- again, never needed that, though I did switch to and fro a few times just to see if the redundancy is there.
My preferred streaming apps are OBS and Webex, though I am also working with Zoom, BigBlueButton and occasionally some Microsoft platforms, especially MS-Teams. Performance (up to 1000 participants) and data privacy is sublime on Webex, so I go with Webex when I can choose myself.
This setup has been going strong and reliably during the entire pandemic.
While this period kept dragging on, I kept adding physical Dante hardware inputs and outputs, so that now I get Dante inputs into the streaming machines from outside sources as well, including a UFX+ running in my recording room, via AES, using a Dante AVIO AES dongle. I also use analog input and output Dante AVIO dongles, and three USB AVIO dongles attached to Windows, Mac and iPad Pro machinery. (I don't really believe in Bluetooth for audio, so I have avoided the AVIO Bluetooth dongle so far.)
Plans for the future include an RME Digiface Dante attached to the streaming machine(s), which would make the setup faster latency-wise (though I am not sure that I need that), but especially it would let me do most Dante routings in Totalmix FX, even some of those that I currently do in Dante Via or Dante Controller. Totalmix FX gives me really total control, and the interface is quicker for things like level control, quick patches, and so on. You can do nearly everything in Dante Via or Controller, but it always takes a bit longer to do during a live gig (level controls, quick re-patching and so on require more clicks in Dante than in Totalmix FX). Of course, the "last mile" to/from the wireless/USB headsets would still have to go through Dante Via, but that is not an issue. I like Dante Via on the last mile (the streaming front) because I know it gets me synchronized with my participants. But for the invisible stuff (the inner workings in the streaming and recording facilities), I prefer to do as much of the mission-critical things as possible with RME devices.
I wish there was an "RME Dante Tool" of the kind of the "RME AVB Tool", because I absolutely have no trust in AVB (too many cooks spoil the broth...), while I am feeling covered and safe with Dante, and Dante is expanding fast into the video-cum-audio market currently. Who knows, maybe RME will fit some of their flagship mic-pre carrying interfaces with Dante connectivity eventually? In the meantime, we can of course use the MADI-to-Dante work-around offered by the Digiface Dante, or by good third-party solutions, like those from Ferrofish.