Haiku may have to “see” the external monitor during its boot phase for it to be recognized.
I presume that there is a keyboard key combination allowing to switch between the notebook screen, the external monitor, and both displays? If so, can the external monitor or both displays be set as the power-on default?
Maybe someone who has attempted to hook a projector to a notebook running Haiku already has a solution as this would be a similar situation. Searching the relevant posts did not yield any which were relevant. There is only an old thread about dual monitor support which might be applicable to your case ( Dual Monitors for Haiku? ).