The real answer is that we likely will never know.
The literary criticism that informs theories such as the Documentary Hypothesis and the Deuteronomistic History cannot support the conclusions of those theories regarding who wrote what when. At best, literary criticism can identify different sources, i.e., literary criticism can be used as a form of linguistic forensics to detect forgery, redactions and interpolations.
Neither can literary criticism provide any support for my conclusions regarding the authorship of the Primary History.
But the anachronisms that I have discovered in the Primary History do clearly determine when the stories of Saul, David and Solomon must have been written originally: after the death of Ptolemy II Philadelphus and before the marriage of Ptolemy V to Cleopatra I. That is, between 246 BCE and 194 BCE.
Further, the parallels between the lives of Alexander, Ptolemy I and Ptolemy II, on the one hand, and Saul, David and Solomon, on the other indicates a hostility towards the Ptolemies and even implies some salacious details regarding their history that are no longer available to us (the parallels are a mirror, albeit one distorted by polemic; still there likely was an Absalom among Ptolemy I's sons, for example).
Still, I cannot say for certain who first wrote the Primary History. Was it indigenous Judeans? Or was it ruling Hellenes? For a variety of reasons, I select the latter, but unless we find new evidence, my conclusion likely will never be confirmed.