then I feel really cheeky having to ask my family if I can bring children who they don't know and aren't THEIR family.
First off, family or not, no one should ever be asking the host of an invite-only party (which a wedding most certainly is!) if one can bring extra guests, and especially children. If a friend or even a relative did not invite my own child to their wedding, I definitely wouldn't ask if it was ok to bring her anyway. (I'm not having a go at you, OP; I understand why you'd do it - but I don't think your DP should put you in that position.)
OK, now I've said that - I do think there are a lot of variables to this issue. How long a couple has been together, whether they live together, whether the kids live there fulltime or how often, how often the kids have met the extended family, the ages of the kids...Frankly, only you can really judge how it is going to be understood by your family. If you think that your family is not being welcoming enough, you should speak to them about it, and hope that they will consider your situation more sensitively. If you think it will be upsetting to your core family (by which I mean DP and his kids and yourself) not to attend this wedding, you should be able to talk to your dad.
I actually don't think it is necessary for you to take the kids to the wedding. You are not taking another child, right? So it is yourself and your DP attending your dad's wedding. That seems perfectly ok with me. Especially given that there is not yet a strong relationship between your dad and the kids.
My own stepkids live with us all the time. My family lives very far away, so this doesn't come up very often, but hypothetically - I wouldn't take my DD to a family wedding or party and not take my stepkids - but I certainly would go to some things with only DH! Especially if we could not afford the extra cost of attending with all three kids. What I would not do is take some of our children but not all.
I can tell you that after 4 years in their lives and 2 years of living as a family, my stepkids (teenagers) do consider my nephews their cousins (they are less inclined to thnk of my siblings as aunties and uncles, which I suppose I understand). My parents say that they have ten grandchildren - they count the DSCs in that number, with no expectation that the DSCs view them as anything approaching "grandparents" - I think that their view is that they are happy to have as much of a bond as the DSCs are comfortable with, and will let them set the terms - which is nice and comfortable for all of us. I love my parents for that! If I thought they didn't view my DSCs as family, I'd want to rectify that, but you can't force it, you can only explain to them how much it would mean to you - and your DH should definitely be sensitive to your family's feelings and limitations, too! It goes both ways.