You are quite right to want to help your girlfriend with the baby. After all, it is your baby too. I can see where the sleep thing is a problem because I was a very heavy sleeper. But as soon as my first baby was born, even the slightest sound from her woke me. But not my husband. And he was a very understanding and helpful father.
Is your baby breastfed? That is much easier on the mother because there is no business about heating bottles in the middle of the night. You just tuck baby in bed with you and everyone gets a good rest.
Luckily for you (and your need for sleep), you do not have hormones that have made you hypersensitive to the needs of a baby. Women's bodies prepare them for caring for an infant. So it is wrong for her to make you feel guilty that you aren't just like her, popping out of bed at the first cries. You and she need to work out a system where you help her, but she can't expect you to automatically be just like her and hear a baby cry in the night.