I am confident my explanation addresses OPs original question, which is whether two different (and consistent) formal systems can disagree about whether a given

I am confident my explanation addresses OPs original question, which is whether two different (and consistent) formal systems can disagree about whether a given TM halts or not (and hence differ about what the value of BB(N) for some N), and the answer is yes they can. The source of that disagreement would come down to two different models of natural numbers where the theory that proves that a given TM halts only models non-standard natural numbers whereas the theory that proves that the given TM does not halt contains a standard model of the naturals. What you're discussing is a completely different matter that while is interesting, does not address OP's issue and is something I already responded to by stating that one must be very careful to differentiate between what is actually true versus what can be proven by a formal system. You are blurring the lines between the two but that distinction is critical to understanding the point being made.