Pippi, if your dh is anything like mine he is seeing this period as a short moment of kicking back before all hell lets loose when dc2 is born!
When I went on maternity leave my dh saw a small window where he could lie-in, drink beer, watch movies till 2am, pretend the house and dd1 did not exist and become almost Homer-esque in his lifestyle.
Admittedly he had been looking after dd2 full time for 3 years and was absolutely knackered!
We both knew that things would change when dd2 was born, and I'm sure your dh is expecting all het to be let loose when your dc arrives.
There is a mismatch in the lead up to the birth. Us mums want to tear the house apart and clean every inch, our dhs and dps want to drink beer and pretend they are at college again.
I don't think what is happening now necessarily means this is what will happen after the birth of your dc. He will simply have to take on mor re dd1.
When dd2 was first born I tried to do everything - bathtime, bedtime for both etc. It quickly became clear that to get things done fast so we could both relax, we needed to separate the tasks. That way there was no crying, kids in bed by 8pm (for an hour at least!) and everyone getting more sleep and less stress.
I'm sure it will work itself out of its own accord, I wouldn't put too much pressure on change now (except gently suggesting that dd1 might like to blow some bubbles with him while she has a bath, or she would like a story etc.).
We used to be completely 50/50 when a childless couple, and children have their own way of forcing you to divide up tasks - even if does often fall into gender lines e.g. dw cooks and dh slopes off to shed....