Ok, I'm coming at this from a completely different angle and its because of my history with ds. What has the outcome been in regards to the heart murmur? Has bub had an echo? I only ask because this rings alarm bells for me as the falling asleep after 5 mins on the boob and being constantly hungry is exactly what ds was doing.
In our case it was a sign of a much more serious heart condition. He was diagnosed with a murmur at discharge as well but we never made the follow up visit as he had been admitted back into hospital by then.
I don't mean to scare you and 9 times out of 10 murmurs in babies are innocent but please pay close attention to breathing issues, fluid retention and fussiness. If she hasn't had an echo, can you ask your paed why not.