I love live chat as a customer too, but from a business point of view, live chat is great if you have a call centre, or staff who never need a lavatory.
Customers won't hang around so if you can't be sure of beginning to service their enquiry within say 15 seconds I'd give it a miss to avoid shooting yourself in the foot.
Would you get away with "not available" in the live chat box? Maybe, but you know your customers and your likely availability better than we do.
You might also be the exception if you have a specialised business where you could block book times, and clients take or leave it because they have the same sort of constraints, but then you're probably into web conferencing territory.