It occured to me that Baron James may have been named after Yaakov Frank.
Jacob "Yaakov" Frank died in the Frankfurt suburb of Offenbach in 1791. Yaakov Rothschild was born in Frankfurt in 1792.
As far as the Rothschilds descending from Shabbatai Tzvi...
If the Rothschilds are from Levites or Cohanim then it would not be possible to be direct male descendants of Shabbatai Tzvi as he was from Judah, being a Davidic descendant. So far I have not seen any evidence of the Rothschilds being Cohanim or Leviyim.