Would there be serious consequences and repercussions if you decided to use your authority and override their decision by getting the engineers to do their job?
I don't think there'd be serious consequences, but you'd probably get a please explain from the fleet office. Although the book says the Commander has 'sole authority over any and all decisions regarding operational control', it also says IOC has the authority for operational control for any commercial issues during the IOC control period, ie before departure.
I would regard PTV problems as a commercial issue, not something that affects the safe operation of the aircraft. If IOC are happy to accept the commercial repercussions (eg. compensation, grumpy pax, etc) of operating a ULH flight without PTVs in F class, then so be it. The Commander could always override IOC, but he or she had better be prepared to justify that decision. The Commander knows what's going on at the aircraft level, but IOC have the bigger picture with respect to the consequences of delaying the flight, eg. aircraft rotations, passenger connections, etc.