Gin, why shouldn't the AP be certificated to Cat 3 limits if the HUD already has that capability. The sensors and engineering should be approved, similarly proof of guidance if other operators have the AP capability.
Is the HUD dual; was it an add on, or built in as standard ?
Don't assume that AP approval will be expensive vs add-on HUD. AP hardware can allocated a failure probability, not so the human. Thus the human certification could require more evaluation.
Who / what monitors the human, how; 'HUD … very capable, hyper accurate', but only if you follow the commands accurately, and who can tell inside the deviation alerts.
There are some very 'interesting' HUD approvals around the world; EASA disliked HUD for operations in extreme conditions (Cat 3); is that still the situation, or do the latest Airbus have dual / dual installations.