Your dsd is a carbon copy of mine it would seem.
I have been a stepmum for over 6 yrs now, I love my dsd and treat her as I treat my biological dd.
DSD still does pull the "I want x or I'm not coming" but she also tantrums for her mum, ie school refusal, refuses to do as she is asked etc. It helped when I found out from her mum that this was happening, it gave me a little bit of relief to know that it wasn't just happening for us.
I found it really hard for the first couple of years, DP would always (and still now to a certain extent) indulge her and his reasoning was pretty much the same as your husbands. Shes only here for a couple of days, she can have what she wants....
That doesn't work in a family environment, it causes resentment between the two of you, your children will resent this as they get old enough to understand and your dsd will learn that she can manipulate daddy into doing her bidding because he is scared of losing contact time.
I think compromise is needed on a few things.
Your dp needs to understand that his actions have consequences for all of you. My dp was horrified when the school called us to say dsd was in trouble over a few things. When they said they were going to call us she replied, "so? My daddy will just say I'm a princess and buy me some sweets..."
Since then he has curbed it a little. Although he still spoils her it is not half as bad as it was, althoug I am braver about putting my foot down as well. I don't ask for toys to be picked up for example, I simply state it as fact. "we are going to tidy up now, dsd you need to do x,y and dd you need to do z,z whilst I do this."
I do think that a two year old can help with putting things away, so you could say "dsd I need you to pick up this and that whilst I help dd to do x and y".
That way, it becomes a family activity rather than one child doing a chore whilst the younger one carries on playing.
Is your dsd's mum playing on this though, I mean, when your dsd says she doesn't want to come is she just going along with it rather than sayingno, you are with your dad at x time and me at y time, that is how it is.
Because we found that at first, dsd's mum was not promoting contact so that contributed to the situation and the whole fear of not seeing dsd for my dp. Once things were more amicable her mum began to say no you are going to your dads this weekend and it got easier. There were boundaries set and a routine was established.