Maybe you could have the live chat button only show up when you're online. The impression it'll make on your customers depends on your target market. I suspect that local customers would not put much weight on 24/7 sales chat, or even 24/7 support.
Can it still serve a purpose?
I've seen a number of companies, some quite big, having a pop-up thingie that invites you to chat with a sales representative after you spend a certain time browsing the site. I think this could be quite effective at clearing up any questions that potential customers might have and take better advantage of impulse buying.