IIRC the ‘shared monitored approach’ had its origins in Trident all weather operations and the then emerging two crew aircraft; the BAC 1-11 becoming Cat 3 capable.
There is a balanced view of some of the arguments for using the approach procedure in
Eliminating "Cockpit-Caused" Accidents.
A problem with HUD is that for true monitoring there should be two HUDs or a system which allows cross checking or internal self-monitoring with a very high integrity level - as required for Cat 3. One of the issues is how to monitor the display guidance, and for manual flight, how accurate the PF is following the command. This can be achieved with separate monitoring of conventional instruments or via the FD, but again the FD integrity requirements and possible need for dissimilar computation and certification proof might involve prohibitive cost – particularly Cat 3B.
Most modern aircraft have flight guidance systems capable of Cat 3 thus the use of the existing autopilot is a much cheaper option (and lower training/currency cost).
Where HUD is available then it could be used to aid monitoring the autopilot guidance (the aircraft flight path), but this can be done with similar accuracy with conventional instruments, and again at lower cost (training). Some concepts explored a mixed autos/HUD operation; IIRC the A320 with HUD may have such capability but
AFAIK it is not used to its full extent - hybrid system.
HUD could aid the head up pilot locate visual cues and the runway, but arguably due to ‘conservative’ minima, few landing decisions would depend on this, and for fail op, it may not be necessary – nice to have but not essential.
Retaining existing operating methods may have greater safety benefits from interoperability where pilots move between fleets, and by reinforcing conventional flight instrument monitoring skills.
BA should not need to justify this, particularly with their proven approach to safety and all weather operations.