M.H.
My husband's Mom seemed to do absolutley everything for him so he was not used to doing stuff around the house. We used to fight about this on and off, when I asked him to do stuff he would say how hard his day was (which most of the time it must have been, his work is very labor intensive). This used to really annoy me though because my day was hard with one kid at home. Then I was expecting my second and was worried how hard it may get.
Basically to stop any arguing I stopped being so nice about it, I didn't ask him to do stuff but told him I needed it doing. So instead of "honey, would you mind doing the....." I started to say "babe, I need you to do this" and he would do it, at first with some complaining but let the complaints go (or eye rolling), do not get into a fight about who puts out the trash, it's not worth it. So I would let him grumble about it but ignored it and carried on telling him I needed stuff doing which he got used to and stopped complaining. I did show my appreciation for what he was doing aswell, I always say thanks when he does something I ask him to do, that shows him I appreciate the help which makes him want to help me more in the future. You could also start asking your daughter to help out by putting her toys away and little things like setting the table or bringing her plate to the sink when she's finished. She is old enough to help with simple tasks now, and that may guilt your boyfriend into helping you more.
If your boyfriend really cares about you He'll get it. My husband became way more supportive after I had our second son. He helps a lot around the house now, I don't ask him to do most of the stuff that needs doing. If he really refuses to help you after the second child is born maybe you need to reconsider your situation, like you say you're doing it alone now anyway so what difference would it be not to have him around? Think about it.