The idea that Zeno questioned the authorship of the letter to Pythocles is speculative at best
Yes I fully agree. Although Usener puts this in the "Spurious Letters" section, ὑποψία only means there was some uncertainty regarding them.
The Greek, indeed, is broken, but it could go something like:
"[Zeno, because he was] approaching the [writings] of the men [i.e., the founders of the school] with precision, regarded those [precise points] that were accepted by them from the foundation [of the school] as very important – therefore he acquired some uncertainty regarding..."
Which makes sense -- and happens to us all the time!