Fiance doesn't want to help at night. Rant

I have been sleep deprived for about 2 months now, never getting more than 4 hours of sleep. My 3 month old daughter has been taking 30 minutes to an hour at most naps. I can't even get to sleep by the time she wakes up. My fiance helps when he gets home around 4 but ends up falling asleep around 8:30-9 pm and doesn't wake up until 5 am for work. I want to let him rest because I know he has hard days at work most of the time but I'm in desperate need for just a few hours of sleep. I just need 2 or 3 hours of sleep on his days off. When he takes care of our daughter after he gets home from work it's never long enough for me to take a little nap. I also do all the house chores. Everytime I bring up the fact that I'm very sleep deprived he either ignores me or shuts his eyes and acts like he's sleeping. I get no sympathy whatsoever and it feels like he doesn't even care that I'm sleep deprived. I've been bringing this up for about a month but nothing has changed. Well last tonight took a turn for the worst. We got in an argument about how he won't help me get a few hours of sleep and he starting packing a bag with his stuff and said he was going to his parents house for the next 5 days and blocking me. He told me that we are not okay and he's not sure if he loves me anymore. I had to beg him to not leave and agree to never complain again about being sleep deprived. I feel so lost, I don't know what to do.