By the looks of things, the only aircraft out of the A320ceo, A320neo, B737NG, B737MAX families that can (or will be able to) operate out of BHD at MTOW are the A318 and A319neo. The runway would need to be extended by about 150-200m for a fully loaded A320neo, and by almost 500m to allow for a fully loaded 737-8MAX.
Looking at the figures, it seems that, in theory at least, it is possible for an A319neo to operate BHD-LCA or BHD-SSH, but an A320neo would require a passenger limit of about 160 (hence making it pointless to use an A320neo over an A319neo). However, the lack of interest from airlines in the A319neo means that it is extremely unlikely that BHD will ever see the introduction of these types of routes.