Does it help you to carry a little snack? I get funny in the morning if I don't eat (although fortunately I don't usually have anything as severe as fainting) and usually carry some food with me just in case. I know that depending on the type of blood work it would have to be fasting but the food is insurance in case you'd feel faint.

Perhaps the doctor's office could draw your blood right away first thing so you can eat your snack immediately afterwards and then be ok for your appointment. I don't know if these suggestions will help but thought it's worth a try.