First of all: More visibility for chat and a general bigger awareness of what chat can be used for would be awesome. But I don't think tag or title based approaches, like mentioned in your question/comments, are going to help here.
From your comment:
How would we differentiate b/w an SO question and a chat room question? Maybe provide a suggestion for the chat room as well, but not as prominently, especially for items that are more discussion-oriented (and might be closed as off topic, etc.).
It's one of the reasons suggesting a chat room when writing a question is probably not so great an idea to get more people to come to the ASP room. It's going to be pretty hard to get an automated tool to recognize question titles as better belonging in chat. And, well... chat isn't really meant for answering those questions that don't fit on the site anyways. From the help center:
When should I visit chat?.
- for real time collaboration
- to meet the fellow members of your community in a more social environment
- for less structured, casual (but still roughly on-topic) conversation
Though some questions might be opinion based enough to start a less-structured, casual but roughly on-topic conversation, it's still different (to me) than having a question dropped in and people answering it: a conversation is a lot more people reacting to other people than just the original question.
Again from your comment:
I hoped we could somehow use SO's built-in tools to do this, but open to suggestions
I think there's one built-in tool that may be used to get more people to your chatroom: comments. It's mostly manual work though, not automated.
Use the people from the room and comments. If you see a question that's not good for the site (closed, or about to be closed) but might be a good topic for discussion in your room, you can always leave a comment. Check if the asker has the necessary 20 points to talk in chat first though, so they can actually participate.
On other questions, you can point out improvements and tactically drop a line inviting the person to chat too for further help on improving their question, if the character limit allows it (again, check for reputation points). You can drop comments on answers/questions if you've been discussing them in chat, and tell people that the chat conversation leads you to suggest a certain improvement or requesting a certain bit of clarification.
They're small ways of rubbing your chatroom under people's noses, telling them that hey, there's this place you may never have seen before that has some awesome people willing to talk to you and help you out!