I'm curious as to what your baby is eating like before falling asleep. Do you hear swallowing? (it sounds like a faint "uh" or "uh-uh"). Do you see rythmic movement in baby's cheeks...usually baby should be opening mouth, pausing briefly to allow milk in, then closing a little to swallow. Are you able to tell when you've let down? Sometimes babies can get "bored" with a slow-flow and fall asleep, but then they can also fall asleep when satisfied.
I agree with oceanmamma to let her just hang out on the breast and that will stimulate milk production. I really hope your consultation goes well and you are doing great!!!