Is there a precedent for a governing party to start a motion of no confidence against itself?

Answer 1

An example of a similar situation you may see in the real world:

A head of state is a member of a governing party. The head of state carries out an action or actions through their office which are widely unpopular, and damage the image and popularity of other legislators in the governing party. Other members of the governing party may, in such a circumstance, choose to vote no confidence against the head of state, despite said head of state ostensibly also being the head of the governing party.