Being diabetic doesn't stop you from having your ears pierced, just means that you've got to be ultra pedantic ensuring checking and clearing etc has they heal to ensure that it doesn't become infected, and any signs that it might be you seek medical advice.
It's probably best to have ears pierced the old fashioned way, with a hoop rather than a stud, as you can easier clean and inspect the wound with hoops than you can will studs..
As to Claire's disclaimer, I suspect this it more to company policy and how they train their staff, but have a look around to see who else offer's ear piercing and would they be happy to carry out the procedure, but do check out their training and council registration etc..
But do ask her DSN about it, as her currant control might be the clincher, if she's regularly runs to high levels, then this could increase the risk of infection, so it maybe that she'd be advise to wait a bit longer until her control is pretty stable so risk of infection can be kept to the mim possible.