It seems to me that the two formal systems can disagree about BB(n) for some n without disagreeing about the state of any given Turing machine at any specific time step. For example, ZFC+CH might non-constructively predict that some machine M halts, while ZFC+notCH might predict that M does not halt. If all machines other than M can be either run to completion or proven not to halt, then the value of BB(n) would be provable in ZFC+notCH but undecidable in ZFC+CH.