Husband saying I'm too needy after baby

Sorry in advance for this being so long.

I'm 5 weeks PP. My husband is currently on week 5 of 12 weeks of paternity leave. I'm exclusively pumping because my baby never latched well and we both want her to get the benefits of breast milk. I had kind of a traumatic birth and my husband was pretty affected by it too. He was really supportive the first week. Then came the outbursts and criticism. He flipped out on me because I got the address for our pediatrician wrong, but it was the address Google gave me. 5 days PP and emotional, he yelled at me in the car with our newborn while I cried and told him I didn't mean to make that mistake. He berated me the whole drive.

Since then he has snapped at me mostly in the middle of the night when we're both exhausted. He'll say things to me like "you never want to take care of her" "you just want to sleep" basically telling me I'm lazy when I ask him for help through the night. Since I'm pumping, we decided he would feed her while I pump at night. I get up to soothe her in between as needed, and stay up with her when she just won't sleep. We take turns with her during the day because she's been fighting naps hard, and he'll tell me to get a nap during the time he has her. It sounds great until he brings it up every night, reminding me that I got a nap and I just left him alone without helping him. I let most of the comments go cause I know he's tired too, but the nap thing bothers me because it's like he's doing something nice and then just makes me feel guilty about it later. The house is clean, baby taken care of, so I shouldn't feel bad about taking a nap. It's the only thing I do for myself.

He's been doing it more often, and it's starting to really weigh on me. Hearing that he thinks I'm not doing enough for our daughter is upsetting when I'm pumping around the clock and doing everything I can for her. We talked about it and he said he realized he messed up and would do better. He says this after every time he does it, but there's been no change.

His biggest complaint is that he doesn't get to work out regularly. We have a home gym but when he's out there he won't hold her or help with her at all and he's there for about 2 hours, then showers, etc. I pump every 3 hours while my supply is still not regulated. All I ask of him is that he helps with the AM feed so I can pump before he works out because our daughter is at that stage where she wants to be held constantly and we have to be standing up moving for her to be happy, so it's a struggle to pump and do that. He claims I don't let him work out because I ask him to work around when I need to pump. He regularly brings this up and I told him we should communicate the times we need to do things so we can both get our stuff done. He thinks he doesn't need to communicate that with me, that I'm telling him he needs my "permission to work out" and he won't listen to me when I try to explain my reasoning.

I tried talking about it today with him and explaining how I feel and he just kept taking the conversation in circles, saying i don't let him have any time to himself, I'm so needy and always need his help with the baby, that I never offer to make him food when he has the baby. He literally told me I don't make him enough sandwiches and act like I don't want to when he asks. He brought up divorce. I told him fine, he can leave. He wouldn't leave and is still here, telling me I make him feel like shit.

I just feel like I just had a baby, I struggled with baby blues badly in the beginning and his criticism didn't help. I need a partner, not someone keeping score of who contributes more and making me feel like shit for napping when I can and pumping to feed our daughter.

Am I being too hard on him? He still games at night and when the baby is sleeping. He could work out but he won't communicate or plan with me so he screws himself on that. His mom comes to cook for us at least once a week, and I do most of the bottle cleaning and the chores I could do while recovering. No I don't cook much and he cooks most meals, but that's the only thing I haven't contributed to as much as him. I can't help but feel guilty and like a bad mom for not doing everything myself and for needing his help, but isn't that what paternity leave is for? I could do everything, but I'm trying to look out for my mental health and take the help while I have it. And being guilt tripped for that help is not exactly great for my mental health.