I know I'm not the first one to bring this up, and that's even more of a reason to consider adding 24/7 chat rooms and not just in matches. The volume of users will be at its peak when matches are going on, so any concerns with volume shouldn't be a problem given it won't be any more than during matches. There are always the spammers and profanity being used, but as in any site there are ways to monitor that as well. Most sites also have multiple chat rooms as opposed to one and just like in match-ups the chat rooms separated by the country which you are from, chat rooms should be separated by countries as well.
I think the best solution is to have 24/7 Chat Rooms with Supporter only, and here is why this can be a great idea:
1. Reduce the volume of users online given you must have Supporter to access it
2. Great for BB's business, buying Supporter membership is about giving you the extra tools that can make you excited to have access to, those excitements also apply to having access to 24/7 chat. If you have only matches going on that have chat in them, then it can only be tempting to buy Supporter to have 24/7 chat access. It's the same situation as reading the Game Manual and wanting to learn more by browsing the Forums but having to buy supporter only to access it.
3. Requirement of credit cards means proportion of adults will be higher as opposed to just free 24/7 chat rooms. You must be 18 or over, at least in most countries, to have debit/credit card. This is not to be discriminating towards age, all ages should be welcomed and you figure some parents would be willing to buy Supporter for their kids. In a way this is most uncomfortable to advice because I mean not to generalize or discriminate age. But I think its fair to say Maturity within chat rooms can remain stronger if adults were around, meaning less likelihood of spamming and jerking profanity.
4. Accessibility of Forums which remains only to those with Supporter, still will remain secure for those with Supporter. Sure someone can constantly send links in chat rooms for certain forums and that can reduce Supporter subscribers, but that won't be an issue since you must subscribe in the first place to have access to chatting.
5. BB would look more developed with a chat rooms. As with any website, the more tools you have, the better perception you have of that website. People will gain more respect for the site when they can see more development.
The bottom line is that with any additional tool would be added, there are concerns, questions raised, and things to consider. This is an advice and I understand from business point of view there are things which should be concerned and address, so this is not my proposal which it has to work in this method. I'm only gesturing an idea which can be mutually beneficial for the users and the company.
One concern might be is people exchanging messages about trades and transfer acquisitions or such where it becomes an advantage and unfair. But considering people privately can communicate with each other without possible monitoring, can public conversation lack more scrutiny.
Hope this helps as it would be nice if chat access was available to everyone 24/7.