perhaps Encore just insn't capable.
True. Encore does not have scripting. That's what I meant. Encore has many limitations because of that, but it was never Scenarist.
Using only Encore, you can include a button that turns the audio off (goes to a second menu like the one I described).
You might be able to use ifoEdit or similar tools to achieve this with an Encore built image. I do not know.