I believe Green Batt Meeting House was also known as Sion or Zion Meeting House.
But that chapel isn't in Green Batt and it is listed as a separate chapel - see TNA catalogue and also https://www.ismayons.com/nonconformists/presbyterian.htm

Has anyone looked through the 1851 religious census*? That can sometimes be informative about the location and denomination of various churches and chapels in a locality.

* Alnwick seems to be HO 129/559