Ogre - Many years back I had to go through the process with my eldest son who was born in PMRAFH Akrotiri. There was no problem getting him a copy in the UK, though I cannot now remember the process.
More recently (say three days ago) I have been lending a hand for one of my daughters who was born a civilian in Dubai. She was first registered by the hospital and later I registered the birth with the British Consul General. We need document the birth and the process has already been started; we have yet the hear anything. What I do feel is that your case is very clearly detailed in the FAQ;s on the GRO web site.
Q1. What records are held for overseas births, marriages and deaths?
Births, marriages and deaths registered by the British Forces and the British Consul or High Commission in the country where they took place, from 1849.
Deaths that occurred in the Boer War and both World Wars.
British Army Regimental records of births, baptisms and marriages dating back to 1761.
Marine Births and Deaths, from 1837, which took place on British registered vessels.
Aircraft Births and Deaths, from 1948, which took place on British registered aircraft.
I have been away from the UK for so long I admit to being a little out of touch with the mindset of government employees, however, reading the above, my feeling is that if they have no record for your case then there is a problem on their side and it should be their responsibility to seek the information they require to make their records complete ! They have a lot more resources and clout to chase wherever they need to close any gaps.
Whatever, I wish you all the best and hope you manage to resolved the issue.