I think teens in step-families are often a bit of an issue.
A lot of teens are difficult to like. They're too old to be cute, and they're demanding, self-centred and a bit annoying.
The reason they survive is because their parents may not like them at times but of course they love them (hopefully). That makes having them in the same house bearable (and of course there's only a few years left until they are hopefully independent and move out). And then amongst the angst and unpleasantness, there are moments when they are nice, decent, interesting human beings and that gives you hope for the future.
But of course this isn't your teen, you don't live with her and you don't love her (at least, not as her parents presumably do). And there's a lot of truth in the saying that 'Hell is other people's children'. Especially when they've been poorly parented, as it sounds like this child has by both mum and dad.
Your DH sounds like a wet blanket. Why has he accepted being relegated to being a breathing ATM in his child's life?
I don't blame you for not wanting to go away with someone else's surly, poorly-parented teen. But equally, if I could have tapped my useless dad for hundreds of pounds of stuff when I was 15 and got away with it, I probably would have.