I"m sorry there's this imbalance in your home!
I'm not going to bash him here. I do want to make a concrete and positive suggestion instead: If he does indeed do his "assigned chores" as you said, maybe it's time for more assignments! Try a family schedule/calendar that is reviewed by you both and prominently posted, and gives him more he has to do. I hope that doesn't sound too basic or childlike, but I know a lot of families, adults as well as the kids, who live by their weekly calendars. Everyone fills iin what they're doing and they sit down together and everyone knows well in advance things like, "I have my exercise class next Saturday and will leave the house at 8:45, so you have Susie on your own from 8:45 until I come home at 11:00." This means he cannot be surprised when you remind him on Friday night before about a class the next day, because you can legitimately say, "It's on the calendar and we discussed it a week ago and again three days ago," etc. Your daugher's activities should all be on there and that allows you to say, "You know, she has music class on Saturday afternoon and you've never been with us -- why don't you come this week with us both and next week you can take her while I..." He may just find he LIKES to spend time with her.
And oh yeah, once the new baby comes, there will be gobs more stuff for everyone to do and he will have to deal with your older child more-- and you need to ask him clearly to do so, not let him think you'll just work two kids into everything you do. So you can use the new baby as a reason to say, "Let's get a weekly calendar going and reassess who is doing what--BEFORE the baby comes."
If it's done with good humor, with reciprocation, and with clarity -- he needs a copy of it at work on his desk! -- it may work. I wouldn't approach it as "you need to be more grateful" right now (though yes, he truly does), but as "let's all get more organized."
And be sure that, since you're a multi-tasker, that he doesn't inadvertently or purposefully take advantage of that. He has gotten used to the fact that you will just handle things, so he doesn't handle them, including your child. He needs to lose that assumption FAST before the baby comes, and you can help him do that by....not handling some things and being clear they're his responsibilities. Again, I'm not saying to get him worked up with accusations and tears, but be businesslike. I know that this approach absolutely does not fulfill your desire for some real understanding and affectionate gratitude, and I'm sorry for that. But in the meantime, until he might agree to some counseling or just realize how much you do, it might buy you some time off.