Well, actually, you just sound tired, stressed and confused.
You said in the OP that you had been looking forward to it - so other than taking your MIL what has changed?
Is there something about your DD's SN's that make dog hair a serious problem or do you just really not like dogs/dog hair etc? If it's not a serious problem for DD then why not just say to your BIL that you want to be able to put DD on the floor so could he possibly make sure the floor is as free of dog hair as possible.
Once again, the floor hasn't changed since you were looking forward to going?!
... and again... I don't know what SN's your DD has, but what would she normally do at home so that she isn't bored? Isn't there anything you can do or take for her - even a DVD and you can move things around so it's not bothering anyone else too much?
...and well, it's tha WAV or nothing isn't it - you can't change your car to suit her/them 
Isn't there anyone who could come and pick her up and take her home? One of the younger family members who would be only too happy to go for a long drive??
I'm going to say it again and I'm going to be even more blunt this time. Tell them exactly what you want them to do, what would HELP you - you are simmering because they don't help - but to be honest, they probably don't have a clue what to do to help. It doesn't sound like you see that much of them, so your DD is bigger/older everytime they see her - they probably don't know what's appropriate, what's helpful and what's interfering, rude and inappropriate... be specific in what you want them to do for you.
Why do the teen cousins ignore her - why has this been allowed to happen? (by the whole family, not just you?).
2shoes - I'm not picking you apart - honest. Just trying to help by separating out the different issues you have here!
Also - I don't care how ready MIL is to leave, either she leaves when you are ready and she stays over and makes other arrangements to get home!!