Is there any way to automatically start an "offline chat" mode when an invisible contact is clicked rather than the default of "Compose Mail"?
There is absolutely no way to do that, as that's the whole point of being "Invisible" - no one knows that you are online. Suppose, things were like as you ask about in this question:
Then, clicking on the invisible user will lead to the Offline Chat and you will know that the user is actually online - thus contradicting the very meaning on invisible.
You can have the offline chat with an invisible user only if the invisible user starts the chat with you when you are online and visible.