I've spent some time investigating this. I wasn't able to get shout working reliably, so I'm not going to go in that direction.
I looked into the old system again, and they seem to have resolved the UI issues I had with it.
Additionally, my upgrade process is much safer as I have been testing all changes locally before making them live. This gives me breathing room in case there are future breaking bugs.
Because of this, I'm going to re-enable it.
I'm going to keep chat.ledstrain.org (zulipchat) as well, as a fallback.