I see you are in Wales. It's no surprise that the relevant info hasn't penetrated that far yet, but its good your doctor is prepared to refer.
I'd say the Andrology clinic in London is a good bet to get the help you need. Not sure if you could be referred to Prof Geoff Hackett, who is closer in West Birmingham, but either would take your issues seriously even on 12.7.
Prof Hackett is at Good Hope Hospital Treatment Centre
Good Hope Hospital, Rectory Road , Sutton Coldfield, West Midlands, B75 7RR
As to your questions:
The UK guidelines state LH and FSH need to be taken in addition to Serum Testosterone and no lab should refuse to do those tests as they define Primary from Secondary Hypogonadism. Others like Free T SHBG and Oestradiol are not essential to the initial diagnosis so would not normally be ordered from a GP's office.
ignoring symptoms is not good doctoring in my view, but in fairness the adherence to numbers dictated by the NHS is nothing new. Education about the facts has not been circulated with sufficient emphasis on how it changes the historic attitudes and approach towards hormones in mens health. Hence my opening comment.
Lets hope he gets the guidelines quick and reads them.
Borderline has two thresholds now. One between 8 and 9 which is borderline clinically low, the other 12 - 13 which is borderline upper treatment threshold.
The target for therapy is a modest 15 nmol/l which is alright for some, like me for whom the halcyon days of youth and middle age are long gone.
The difference between going private and NHS in situations like this is time. Getting an appointment with either NHS options will take time, although it can be shortened by a phone call to the relevant office begging for an early appointment or saying you could take a cancellation.
It's not something anyone can advise on really because you need to weigh up the pros and cons yourself and how it affects you.
If your GP would be prepared to accept the word of doc Savage and prescribe thereafter, it is clearly an option to consider.
If you are prepared to wait it out to get the help on the NHS whilst educating your GP so others can benefit down the road, that is your choice.
I hope it helps,
You have all the info from Matt and myself so the choice is yours.