Usually, there will be so many changes made in the family incident or character that it will be unrecognizable. Secondly, they won't notice until the story sells, and by that time they will be so overjoyed that you haven't been wasting your time that all will be forgiven. And if you're still worried, circulate a one-page summary of the story that indicates how you've changed the family story and see if anyone objects. You should be able to work around this sort of thing. However, if the family objects strenuously, either divorce them or cave in, depending on which is more important to you.
Most families are kinda proud of their black sheep or horrid experiences.
If this were the case, it is understandable that the family would wish for this story not to get out. This usually shows as inequality or contrast between the main characters. Sling Blade, for example, features a guy who suffered through a monstrous childhood and who ends up hacking his mother and her boyfriend to death. For this he is committed to a mental hospital until, decades later, he is released from the nervous hospital. Before long, this friendless wretch, a latter day version of Frankenstein, is befriended by a fatherless boy.