Found something of a workaround: DragonMobile allows you to specify SKEarcon
s, which are audio files that play whenever recording is started, stopped or canceled. I'm going to record VoiceOver making the announcements that I need and then use these recordings as the earcons, so that it will sound like the rest of VoiceOver.
According to a Nuance technical rep I just spoke to, DragonMobile does indeed take over the audio layer and suppress any output during recording, and they don't expose any way around this other than the earcons.