The solution is that David and Solomon did not live in either Iron Age I or Iron Age II. They lived in the Late Bronze Age, which was richer than both. As Immanuel Velikovshy, John Bimson and David Rohl point out, the Late Bronze Age was probably placed too early, due to faulty chronologies and bad carbon-14 readings. Just move the Late Bronze Age a few centuries closer to us, and bingo! Problem solved.
In addition, as dating systems go, the division into “ages” isn’t one. :’)