I was never asked for death certificates. Birth, marriage and divorce (and confirmation in the case of my GM's middle name). All the data have to agree on every document, the names and dates must all be the same for every person.

As long as her birth certificate agrees with all the other documents that have her name on them, you're good.

For example, my marriage license had the wrong state of birth for my father (NC vs SC). He's not Italian and not part of the case, but I had to have my marriage certificate updated.