Hello wafflefish. I’m sorry things are difficult. Here are some ideas. Some may fit your situation, some may not. As always, pick and choose the advice that feels right to you.
1. Value your work at home (caring for the baby) as important. Let’s say your husband is at work for 55 hours a week counting meals and transportation. Meanwhile you’re at home doing equally important work. So far so good -- you’re sharing duties. Now what about the rest of the week? Your husband cleans the cat box and does some of the cooking and dishes, and you do... everything else? Is your husband getting a couple hours of free time every day while you take care of the baby and do chores? Why?
Think of the time from when he leaves to when he gets back as the “busy time” because you’re both doing important work and not getting a lot of breaks. The rest of the time, you should be splitting the work more or less equally, sharing chores and each getting breaks. (Again, this comment may not exactly fit your situation. The key idea is to value the work you do and stick up for yourself.)
2. If your husband consistently refuses to do something that you think he should do, talk with him about it. Make sure he understands the consequences of his decision (like, you’re overworked to the point of feeling crazy, you feel taken advantage of). Ask him what his reasons are. If his reasons don’t make sense to you, say so -- try to work together to get on the same page. If your husband isn’t willing to talk, or if the two of you don’t reach a resolution that you can live with, get additional help. (You say you can’t afford a babysitter, so I’m guessing a paid marriage counselor is out, but sometimes local governments have free mediation services, or if you talk with enough people somebody will know of a good helpful resource. You say you don't have friends locally, but do you know any friendly people who seem like they could help you deal with problems or find support groups?)
3. What kind of relationship does your husband want to have with this child, in the long run? What are his dreams? If he wants a good relationship, you can appeal to that as far as child care is concerned. The best way to have that good relationship later is to dive in now.
It can be intimidating for some people. If he isn’t the type who has fun with other people’s babies, maybe he thinks he can’t have a rapport with his own baby until she’s ___ years old. Maybe he doesn’t realize that if he spends time with her every day, they’ll get to know each other better (Like, what does it mean when she gets that particular look on her face?) and they'll have more fun together.
You say, “She doesn't like Dad much and I feel like he doesn't try very hard.” Can you talk with him about that? Ask him whether he’s uncomfortable with her or doesn’t know what to do? Part of it may just come from putting in the time, smiling, and responding to her in some way.
4. As a practical matter, it will probably help if you assist your husband with learning the ropes of baby care. But if he has trouble figuring something out, don’t let him keep dumping it back in your lap -- because as long as that’s an option, it’s too easy for him to think of it as your problem. If he doesn’t know how to diaper the baby or dress the baby or play with the baby, he needs to take responsibility and keep working at it until he figures it out.
My suggestions here are easier said than done, but I hope they’ll help. Good luck!