I can't find the API which can set this value. I have found this so far:
In the LiveBroadcasts object, there is a liveChatId field which points to the chat. It's also pointed to by the activeLiveChatId field in the video object.
You can query this liveChatId, but the query merely returns a list of LiveChatMessages objects which are the individual lines of the chat, not the SETTINGS of the chat.
The documentation doesn't show any way to update the "live chat" itself, only how to insert and delete messages and moderators to the chat.
Newly created videos seem to have the Live Chat Replay feature disabled by default. See if you can toggle this on.
Look for other stuff that you might want to set in here, like live reactions and stuff.