"She was only conceived without sin through the application of Christ's sacrifice."
Not sure I follow this at all. Do you mean she was physically conceived by her parents in sinlessness?
It's important to understand what the doctrine of the Immaculate Conception is and what it is not. Some people think the term refers to Christ's conception in Mary's womb without the intervention of a human father; but that is the Virgin Birth. Others think the Immaculate Conception means Mary was conceived "by the power of the Holy Spirit," in the way Jesus was, but that, too, is incorrect. The Immaculate Conception means that Mary, whose conception was brought about the normal way, was conceived without original sin or its stain-that's what "immaculate" means: without stain. The essence of original sin consists in the deprivation of sanctifying grace, and its stain is a corrupt nature. Mary was preserved from these defects by God's grace; from the first instant of her existence she was in the state of sanctifying grace and was free from the corrupt nature original sin brings.
Mind you, this was not made up by Pius IX in 1854. He merely reiterated and formalized the teaching which went back to the very early centuries of the Church.
Was it necessary for Mary to be without sin in order for her to give birth to Our Lord? Of course not; God could have accomplished His Will in whatever way He chose. We simply believe that this is the particular way He chose.