If I was the donor and my mother was doing this I would be absolutely mortified!!! This woman has no right!!!! I am very close to my mother and I know if I explained to her that I have NO ties to this child legally or emotionally and that the child will NOT be playing the role of "daughter" or "granddaughter" in our lives, then she might get it. The aunt is setting herself up for a world of hurt and needs to be warned by her own daughter!!!!!
Otherwise, I am afraid that your friend is going to have to cut ties with her aunt and cousins. This has become a situation of her child and her well-being vs. keeping a selfish aunt in the family. We all know what the answer is to that one.
This was one of the big reasons why I did not want to use a donor who was a relative or friend. You might be able to trust the donor but you cannot trust anyone else with the information. It was just something I wasn't willing to compromise, and it was worth the piece of mind and extra money. Although, I really think situation is the exception to the rule.
To give the aunt the benefit of the doubt, I will say here that a lot of people really don't understand the whole donor egg thing; especially, our mothers' generation who got PG on their honeymoons. If her cousin explained to her how the cousin feels about giving her eggs to her cousin (like loaning a neighbor a tool) and that the aunt's actions are causing tremendous anxiety and hurt and they are risking the family bond over a child who will have a distant role in her life, then she might just get it and back off. This is how I would handle it without severing ties quite, yet. She needs to hear this from her daughter.
Best of luck!!! I am so heartbroken for your friend. This is sooo tough on all of us!!!