I would suggest having a system in which each board authenticates it's own users, but trust each other using certificates. A bit like claims based authentication.
Would be something like this: http://www.sharepoint2007security.com/blog/2009-10-20/introduction-claims-based-authentication-sharepoint-2010-plain-english
I hope it is easily possible using phpBB, but i fear it needs some custom programming as it is clearly an integration issue.
What you also might do is separating the board and chat databases with a database for dhamma and dharma wheel each and one database for the chat where all the user data from the two boards is replicated to. This still raises the issue though that you would need to login separately for the board and the chat, but to solve this you really need an authentication and authorization service for both boards (bit like single sign-on
Unless you have a way to completely go web-service based with authentication and authorization on phpBB or have time and funds to let a bunch of programmers do this, merging the chat would be technically infeasible IMHO. PhpBB and Ajax Chat are made to be a simple websites with a database behind and not enterprise applications.
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