You know, Lance...I was thinking about this last night and you DO have the ability to set a Welcome message, by Department, in Live Chat. See the screenshot:
You can set up a Welcome Message for each department you have set up for Live Chat, as well as the "waiting" message the customer sees prior to reaching an agent, and even the "Agent Welcome" message that appears once a chat is handed off to an agent in that Department. I think either of those, or all of them, would fill your need.
To get to this area, log in as an Admin, then go to Settings and under Configuration, click Live Chat. Select and edit a Department, and when the modal appears, click the Custom Messages tab. Here you'll see your options. I think that will work out for you.