Your Dr only needs to tell the cops what is relevant to your condition and his/her
professional opinion on your suitability to possess firearms.
Bear in mind that you wont have been the only one in your force area who has suffered from anxiety and so the cops will apply the same 'judgement' to you as they will have applied to others but also considering your own circumstances. If your Dr is anti shooting then the cops are likely to know this too as in the past they will have approached your Dr before for a suitability professional opinion and if he/she was an anti then they would have refused.
The cops are duty bound to gain a professional opinion from a qualified professional, if not the FME (Forensic Medical Examiner)can provide an opinion after you consenting for them to access your medical records. In short they have to get a professional opinion from a suitably qualified person, so yes, it can go round a bit...That is actually a good thing!
If you dont get your SGC first time, dont worry, what
might happen is that they will ask you to withdraw your application and re-apply in X years. That way, if they refuse you it will go against you, if you withdraw the application your next will be your first
I had to wait 2 years from my first application before I could be granted a SGC/FAC, not through mental illness I hasten to add, but then when I re-applied the applications sailed through