There is more to say here, but I’ll leave that to another time and place. For now I’ll just note this part of the BHK Interpretation that you point out:
It is not, in general, possible for a logical system to have a formal negation operator such that there is a proof of “not” P exactly when there isn’t a proof of P; see Gödel’s incompleteness theorems.
Is in some tension with your last statement that the constructive reading of sutta in question has the Buddha affirming that, "the born can not be proven.” Rather, I think it has the Buddha saying something like, “Assuming the born leads to an illogical conclusion” if we interpret the sutta in question as containing an attempt at a logical proof of the unborn in constructive logic. As I said before I think that isn’t the only possible interpretation.
However, I do think there is an unambiguous case where the Buddha makes a logical proof - not just stating inferences - in the Pali canon where if you take make a constructive reading something like the above is the result, but again another time and place. 