The issue you have is that the vast majority of Beamish trade comes from direct car and coach trips. There likely isn't the demand for a direct service from Newcastle. Even a taxi for a tourist is relatively inexpensive.
Re the above suggestions. I'm not sure Newcastle - Durham needs a tourist link. Hop on the very regular and cheap enough train to Central and the Q1 to Quayside, by which point the T1 is sitting in traffic
RE: Go North East Service Suggestions