First of all, I must say that miniBB is not class-based. I would not recommend to use it as a standalone class application or achieve something like that. It's not worth. Let's not forget that any 3rd party software should be opened for easy upgrades. If you tweak it too much, you will use an insecure software for the whole life. That's not the approach we propagate on miniBB.
Additionally to that, I must say that while working for about 15 years in Web industry, I have concluded that "classing" is purely academical approach, which doesn't have anything common to the practice. Classing is nightmare. While in the functional code you can easily follow the logic and restore the points, even if it's not commented, Classing makes the code totally unreadable and understood only for its author. Also, it makes the code dramatically slow. Just for letting you know, that I dislike Classing. But that doesn't mean miniBB can't be integrated into the object-oriented software. Bridge to the Wordpress available from Downloads, proves the otherwise.
miniBB is a standalone app, and it should run as a standalone app. You can adjust miniBB to use your own cookies/session, using customized authorization functions, or tweak it to fit your graphical layout. In both cases, it can be done non-destructive way using only miniBB functions, and not your own functions. You also can't achieve good results in speed and efficiency, if you will "class" miniBB the way you suggest.
Search forums already may give you some tips:Using existing site username/password session variablesIntegration with external DB and Session management