You could probably just output OBS to a virtual webcam and just do a regular video call over Discord.
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Did you try using the browser? It's been a few years, but that worked fine for me.
I've been using clients for years without ban but you're absolutely right that it does risk TOS, my day of ban could very well be tomorrow.
The only other way I can think of is using OBS to livestream to an unlisted YouTube stream and share the link to those you want to watch, no one else can access it without the link. I'm not sure how bad the latency or delay is on this, but I think you can tweak some things to make it more low latency
Edit: realized matrix via element screen sharing doesn't have audio sharing either so I removed it
For what it's worth, I've been running alternative Discord clients for years (Webcord, discord-screenaudio, and now Vesktop/Vencord) and haven't encountered any issues or bans. By far, the most polished and well integrated is Vesktop/Vencord. I don't consider my Discord account worth risking either, but given that I've yet to see a verifiable report of someone losing access to Discord for using an alternate client (even the ones that enable Nitro subscription features), I think I'm pretty safe.
Personally, I'd say risk it for the biscuit. There are some hacky workarounds but all of them are annoying to set up and finicky. As for alternative platforms, I'm not sure...
I use Moonlight & Sunshine for streaming. It works really well, but it needs a lot of bandwith when you stream to more than one person.