You don’t need your mother’s blessing to go forward, but it would help to explain that it is her issue — and that there isn’t much you can do to help her with it. It sounds like your parents have your best interest at heart, but don’t really understand that it pushes you further away when they act like this.
Explain to your parents that you are an adult, and you need them to appreciate that. Let them know that approval of your girlfriend is not a requirement for you dating. Don’t try to fix this. Don’t try to be the go-between between your mother and your girlfriend. Focus on your relationship with your girlfriend and keep putting the responsibility for your mom’s feelings back on her. Don’t live your life trying to please your mother.