How old are you? (You can give a range if you do not want to post an actual age. Under 14, Under 17?)
Sometimes adults act stupid. Period.
Your father's girlfriend has her life. She sees you and your sister as intruders in her life. Add that your father does not have the ability to make you part of his life with it being a hassle to her.
She is just having a snit fit.
She does not hate YOU. She hates what you represent. Huge difference. A difference that is hard to feel is different, I realize. I have been there. My father's second wife seemed to hate us.
Can your father make other arrangements for you to get there? Can your mother drop you off and your father finds a way to get you home?
Here is your challenge. Be a bigger person than the adults here. Smile and be nice to your father's girl friend and her kids. You will know in the end, no matter what happens, you did the right thing.