What "school of thought" did the Jews of Mecca (or Bakkah) follow that time? Did that school of thought have any major differences with other schools of thoughts (that time or today)? Provide some if possible. If this is impossible to answer, what about trying to speculate: If one collects the information about (some) Jewish "beliefs" which exists in the Quran and other reports, and then analyses it (it has to be interpreted correctly too though), one might draw a conclusion or pinpoint that the result given in the analysis is similar to sect X's understanding or some schools of thought have inclined to similar understandings.
According to Dr. Yisrael Ben Ze'ev in his book "The Jews in Arabia" (היהודים בערב), pg. 111, note 13, there was no Jewish community in Mecca during the time of Muhammad, most likely because it was way too much of a big center of idolatry. It is possible that before it became that, there were Jews there. In any case, during Muhammad's time, there may have only been occasional Jewish merchants who passed by the place. It is possible that he received some knowledge of Judaism from them, as well as perhaps from local Christian Arabs, but most of his knowledge of Judaism came from his interactions with the Jews of Medina.