Okay, after some playing around with other virtual soundcards (Voicemeeter), I landed on one with a buffer combination that works (Voicemeeter VAIO). BAE ignores VAC-based virtual cards like Breakaway Pipeline, which is what made me think it was not intended to do this. So download Voicemeeter Banana to get the driver.
Input: KS, “Breakaway Pipeline 1” (duh) 480, 4, 48000, 2.
Output: KS, “VoiceMeter vaio”, 240, 4, 48000, 2.
Audio Realtime Priority and Adaptive SRC enabled.
I did not open a DAW and try to record from it, but I’m able to listen to it by using “Listen to this device” and it doesn’t have any noticeable latency. There’s the occasional pop, so the buffers don’t seem perfect yet.