We don’t discriminate between doors open/shut - our point of dispatch is “moving under own power” (Authority restriction), although the aircraft is actually certified for such after engine start.
Practically, you could look in the DDG and see what the consequences of the particular fault were and any engineering action that might be needed. If the answer was “none” and “none”, apart from an entry in the tech. log, then I would be tempted to “discover” this fault post-dispatch when operating under the QRH...