As I was looking for names possibly derived from Bejarano through Bedzerano or Bidzherano, I discovered Bizaron and Bitzaron, used as the name of a location in Israel. I wondered if the place could have been founded by a Sephardic member of the Bejarano clan. However, it turned out that the word is Hebrew and means Fortress.
This in turn may have influenced some members of the Bejarano clan to change their name so that it would be closer to Bitzaron. Meaning that they would have expected their name was derived from Bitzaron rather than Bejarano. It does not hurt to speculate.