I came up with an answer of sorts to my own question.
I downloaded the free trial of Audio Hijack Pro, and set it up to "hijack" the audio output from Final Cut Pro. I didn't bother recording it, just muted it, and I was able to continue working (i.e., watching youtubes) while I captured.
I don't love it, because putting third-party stuff on my edit system gives me hives, but it worked.