If memory serves, Franklin's son headed some kind of royalist commission to try (in all but name) Bennie baby for some transgression (may have been something about circulating private letters, can't recall). They had been estranged even before that time.
You are confusing separate incidents.
Franklin somehow wound up in possession of private letters of the governor of MA, which he forwarded (or maybe copies) to his political opponents. Not terribly ethical for a man who was the postmaster general of the colonies.
His son was not involved.
Their estrangement was caused almost entirely by his son taking the opposite side in the Revolution, which BF saw as a personal betrayal.