I had the same problem when I wanted to do a Medichecks test. I took all the stuff with me, the vial, the bits and bobs, and the instructions and information from Medicare. It wasn't a phlebotomist, it was my regular and lovely nurse. She would have done it, but went away to check if she could. In the end she had been told no. It was breaking the insurance rules should something go wrong. She rooted about and found me a local private GP who she said would do it for me. I didn't have chance to follow that up because I had a funny turn a few days later and ended up in hospital. It was then Christmas and by New Year I had lost the motivation to do it.
I should add that I have no trouble getting any normal test I ask for at the surgery, all the cholesterol, liver, kidney and full blood counts are routine anyway, but if I want extra tests for vitamins and minerals, I get them.