P.P.
My husband and I also work opposite schedules. I have had similar experiences that you describe where I was worried about leaving my baby at home with my husband because he was so tired. (He's also a police officer so I worry about him being too tired when he's working 4pm-2am.) It caused a lot of problems during our first child with me resenting him and being angry at him. It escalated to a point that we almost called it quits. After I had my second child, I realized that he isn't sleeping on purpose. It really is extremely difficult for him to stay awake when he's completely exhausted. It didn't mean that he doesn't love our children or that he is putting his needs ahead of the children, it just means that he is simply too tired to be able to adequately care for the kids.
So I quit blaming and started working out solutions. If you have the option of taking the child to the sitter in the morning, I say do it. Let your husband sleep in on the mornings that he seems incapable of watching the child and leave him a note telling him what you did. Any feelings of bruised ego will be overshadowed by the relief at being able to catch up on his sleep. But then you will also have to sit down with him and talk to him rationally about his sleep. Don't be blaming or critical. Just be matter-of-fact... tell him he's a great father but he doesn't appear to be getting enough sleep and then figure out together what can be done to help him with that. Point out that as the baby gets more mobile, she can get into a lot of harm and you know he would feel awful if she got hurt because he wasn't able to wake himself up enough to adequately take care of her.
One last word of advice in case you have a tendency to do this... I know I did with my first child... Don't keep a running tally of your hours of sleep versus his hours of sleep. It isn't worth the resentment and anger it may cause. Just accept that you are both tired but you will reach a point when your daughter is sleeping more and is more independent and you will have more opportunities to sleep. I read some advice after I was pregnant with my second child that I wish I had when I had my first:
You can be tired, or you can be tired and angry. Either way you will be tired, so choose NOT to be angry.
Good luck. And be sure to tell your husband that he needs to let you have a chance to sleep in once in a while too!