If you install BlastChat client module, it will show others that you are inside chat. Upcoming client 2.3 is in final testing stage and features are frozen.
It is possible even now to create new module (your own), which will allow you to invite user to chat (not from inside chat but from your website) and give link roomid so user will be navigated directly to particular room (will work with client 2.3)
For invitations sent directly from inside chat, we would need to add functionality inside client as well as inside module (may be in next release).
you can add them inside chat by using the place where you can add banners and ads. You can put your own scripts there