Yes, "get_tree().get_current_scene().rebuild_slave_list()" will do exactly as you expect.
To prevent spamming the name change, I'd probably add a member variable to person.gd to track which day the name was changed and then essentially put the stat effects on cooldown. Though the simpler method might be to simply subtract some energy from the player as a minor cost and require sufficient energy to enable the button, which doesn't fit quite as well intuitively but works okay as a game mechanic.
"Mute" gets attention because it's obvious, but there's always room for slaves to be more expressive in terms of their traits.