Hi there,
I can build your chat application with PHP, MySQL, HTML5, Jquery and Booststrap.
I created similar a couple of live chat applications before, one integrated in a website and another as an iframe (anonymous chat) that could be used on any site. But I guess you need it integrated with your users and login / registration scripts.
Responsive design / layout, message refresh via Ajax calls, MySQL storage and I can build an admin back end for you to manage chat messages.
Custom build, simple, fast and responsive.
I am ready to start this weekend, message me if interested.
Regards,
Calin.