If you establish BpS as a strata in your spatial model (usually the primary strata), then every state on your landscape is uniquely identified by its stratum and state class. Then sharing names of states across strata is not at all a problem (in fact it is a very common thing to do!).
Bottom line is you do not need to have unique state class names across BpS. Just make sure you setup your strata correctly. Exercise 3 in our new self-directed training course covers this if you are able to gain access to it!