Tell me about it mate, even if you go in with Symptoms, you have to beg for your life for them to check just testosterone, which costs them £7.
I'm not sure where you are based, there isn't a ton of choice and it isn't cheap, but i'll tell you the options I've found. If you live near manchester or London there are a few things worth knowing about, otherwise, do you live near a spire or nuffield private hospital? There is a company blue horizon medicals that allows you to get private bloodwork through them, they act as your GP so your NHS GP has nothing to do with it and it doesn't go on record etc, although the results are legit so if there is a problem you can bring the results to your GP. I've not used the service yet but I plan to soon as ive been doing self prescribed clomid.
Anyway for Nuffield the website is privatebloodtestsDOTcom
For spire it is privatebloodtestsDOTco.uk
Pricing is different for each, and they tend to have different profiles. the websites are a bit **** but what can you do. Nuffield has a Investigation of Hirsutism profile, which even though aimed at women, has LH; FSH; PROL; Testosterone; SHBG; FAI; DHEAS which is all the stuff you'd want
Its all about how much you are willing to spend really. I don't see why your GP won't test you for basic stuff like full blood count, thyroid, kidney, liver, Vitamin B12, Folate, Vitamin D etc