I had my suspicions the lmlk bullae dated to the time of Josiah (on a number of grounds). I also had my suspicions the Rosette system dated to late in the reign of Mannaseh. I now think I have reconciled these ideas and, thus, I can suggest the chief reason no stamped jar-handles date to Josiah’s reign (at least, in my view): they were replaced by the lmlk bullae! Barkay’s explanation for their purpose seems to me satisfactory. I now suspect that, with further careful sifting, more lmlk bullae will be found at Jerusalem (where I suspect all the unprovenanced lmlk bullae were found). I should also make a map of places mentioned in the bullae and a chart of their distribution among Josiah’s regnal years.
UPDATE: According to Robert Deutsch, the Hezekiah bullae, the fiscal bullae, hundreds of other bullae, and a bunch of lmlk stamp impressions (!!%&$#&%&!!!!) were found by looters at Keilah. Apparently, my suspicion of all the lmlk bullae being found at Jerusalem was wrong. I still say any Bethel bullae should be dated to the time of Josiah. Was Keilah the administrative capital of the Mareshah district?