I went through the wringer trying to figure this out. The easiest solution in the end (but also most expensive) is what Adam (and a producer friend of mine) recommended: Loopback by Rogue Amoeba.
https://ibb.co/5Fn77g6
1. All I did was add my Babyface (with its "Loopback" feature activated on the master out) in the very user friendly GUI of Rogue Amoeba's app (also coincidentally called "Loopback"; screenshot uploaded to url above).
2. I then chose my Babyface sound card as usual in Mac OS Sound Preferences.
3. I also chose the Babyface in Traktor preferences ==> Audio Setup.
4. But I chose Loopback Audio (Rogue Amoeba's application) in OBS after hitting the '+' sign and choosing Audio Input Capture.
It just worked. I'm able to monitor headphones through the Babyface and I tested the broadcast in Twitch.
The downside?
It's $99 US, and you only get 20 minutes on the trial version before it starts adding white noise. So you pretty much have to buy it. Some friends of mine have had luck with other apps (Blackhole, iShowU, Jack), but the one that worked for me is Loopback.
It took a lot of digging to find this solution, so I hope it helps others in a similar position.
A.
Edit: I did this with my video camera off. When I had it on, there was some weird artifacts on Twitch while listening on my phone but that's most likely either computer or internet related.