The short answer is you are not unreasonable (at all) for suggesting he get his own prescription.
The long answer is that diazepam is a relatively inexpensive drug, so if you can afford to fly, you can afford a small number of tablets. It is also a highly controlled drug, so to be found in the possession of it when the prescription is not his own may change the face of your trip entirely (and result in controlled substance possession charges, and potentially conviction).
I have worked both in law enforcement and mental health - while it is more convenient to just take his DM's, it isn't wise.
From a mental health standpoint, fear of flying is just about the most common phobia in existence - there is no shame in asking for something from a doctor to make it safer and more tolerable. His GP will know if there is anything in his medical history that makes diazepam a poor choice (respiratory or cardiac issues, for example) and offer him an alternative. If it is a safe choice, an appropriate dose prescription will be offered in the amount of tablets he requires.
Anot earlier poster mentioned that her GP didn't do any "checks". This is not the case, any prescription is considered carefully against the patient's history, other medications, current complaint etc. Just because the physician does not spell out every step does not mean it is not considered.
From a legal standpoint, the abuse of prescription drugs is becoming a larger and larger problem, and as such, vigilance and prosecution is increasing in accordance.
Do you honestly want "convenience" to run to chance of destroying his future?
Please, have him see his GP - you are absolutely in the right - it isn't worth it...