I'm sorry for your loss, but I agree with others, in that I think your grief is stirring up anger which is easier to channel. Of course you're upset and it's not what you would have wanted, but when your family accepted the new girlfriend, you were accepting that your father had found a new confidente and pillar of support. They've been together for 3 years. I think it quite natural that he would want her at his wife's funeral.
Her involvement in the planning of the funeral, whilst frustrating and feeling insensitive to you, may be her way of trying to show your father that she understands that your mother's passing is a huge emotional upheaval, and may have been her attempt to show your mother the respect she deserves. Some new partners would have wanted nothing to do with it, because with the spouse gone, they become the focus. From what you've said, she didn't try to make herself the focus, but made your parents' marriage the focal point of the day.
Your feelings are understandable, but you're looking at things from a grieving daughter's point of view.
My grandfather buried his wife a few years ago. He was completely broken by it. Travelled in the hearse with his wife to the church, and took his dog in the car with him. Held his dog in his arms throughout the service, burial, and wake. Since then he has shut himself away and refused to see anyone. His only source of comfort is his dog. They were together for 60 years, and the only way he can cope with what has happened is to cling to a source of comfort who asks for no sympathy in return. A grieving child does. A friend or relative does. The dog doesn't, and neither does your father's new partner, because she is not grieving herself. 