You face a steep uphill climb there! It seems that there are vital records entries for her marriage and death (plus a gravestone that documents her death date and age), but no such record for her birth or baptism.
I assume that the Mayflower Society is challenging you to find a will, a land record, a contract, or similar document that will connect Temperance to her parents (or at least whichever of her parents is a proven Mayflower descendant). Do you know whether descendants of any of her siblings have qualified for membership? If so, do you know what records they found and how they found them?