An update - I think I've inadvertently fixed the issue.
My radio did not need a lot of drive where I was feeding in my audio. About 70mV @ 1KHz to deviate 3KHz. As a result, the TX pot on the VOTER board was too course when setting the deviation level. I inserted a 4:1 voltage divider to give the pot more resolution. After setting my deviation level, I noticed that the "pulse" noise had almost disappeared on my service monitor (no de-emph). On my radio and with the service monitor (de-emph) it had disappeared.... so it appears that having the voter board TX level set low means that noise can be picked up and put on the audio output.

Now to move onto the next issue :)