What the nurse is really saying is they send the blood off to a laboratory. The lab cant use a fingerprick sized sample. Good luck finding someone with a machine.
BUT I really doubt taking a sample from a vein will be an issue. You had a cannula or tube, not a needle left in and either the drugs irritated the vein or you developed an infection (phlebitis). We normally replace cannulas every 24 hours in the UK to avoid infection. It is very common to get a hard, red or painful vein - probably 2% after 48 hours. I do not bother investigating it. We simply treat it with pain killers and topical anti inflammatory cream plus antibiotics if the skin is also infected.
The risk of you developing a problem giving a blood sample is therefore the same as for anyone else - tiny. You may get a bruise but infection or anything else is highly unlikely. Trying to change your member state is disproportionate