From PANS-OPS (8168)
The pilot-in-command is responsible for the safety of the operation and the safety of the aeroplane and of all persons on
board during flight time (Annex 6, 4.5.1). This includes responsibility for obstacle clearance, except when an IFR flight
is being vectored by radar.
(Emphasis added by me)
Also from PANS OPS (8168)
The adequacy of terrain clearance during any phase of a flight may be maintained in any of several ways,
depending upon the facilities available in a particular area. The recommended methods in the order of
preference are:
1) the use of current QNH reports from an adequate network of QNH reporting stations;
.....
(Emphasis added by me)
So it is recommended to use the current QNH. The meaning of "adequate network" can be debated, but clearly for ATC it means that if they provide you with QNH, the altitude clearances they issue are related to that QNH.
PANS ATM:
VERIFICATION OF ACCURACY OF MODE C-DERIVED LEVEL INFORMATION
....
3. If the displayed level information is not within the approved tolerance value or when a discrepancy in excess of the approved tolerance value is detected subsequent to verification, the pilot shall be advised accordingly and requested to check the pressure setting and confirm the aircraft's level.
...
Maybe there is strictly speaking no obligation to use the QNH for the airspace you are in, but just choosing the QNH you like for that day isn't exactly a demonstration of good airmanship, nor being cooperative with ATC.
I hope you manage to talk some sense into the FOM. Good luck.