Thanks for taking the time to make the video! This did work with some notes for future readers:
The material needs to be a Standard material, not Nodes. The image is loaded in the color slot of the material and in the Shader Field the channel is set to Color and the texture is loaded via the Material Tag and NOT directly as a bitmap.
Rotating the Projection does work.
Using a texture tag from another object is not a workaround, it's the exact workflow I described.
I can seem to move both texture and cloner without issue.
Shader_Projection.mp4