First, I don't understand why you need BOTH Joomla and Drupal. So far I know these systems are both CMS, and you need to choose only one product.
miniBB then could be integrated with the system you choose (i.e. it will use the existing members table). Starting cost for such bridge comes from our standard price list
and is US $55. It could cost more, if it will take more time to work on it, however despite I never studied nor Drupal nor Joomla authorization algorithms, I hope they are similar to what other systems provide.