I am not a theologian nor do I play one on the internet. With that disclaimer I say that no unborn human commits any sin.
We are all theologians on some level, for we all have "...a system of religious beliefs or ideas..."
That said, scripturally we are conceived in sin, and later born into sin as well.
Psalm 51:5 Surely I was sinful at birth, sinful from the time my mother conceived me.
We are conceived into condemnation. We are by nature, objects of God's Wrath.
John 3:18 He that believeth on him is not condemned: but he that believeth not is condemned already, because he hath not believed in the name of the only begotten Son of God.
While your proposition that an unborn babe cannot sin is possibly true, it makes no matter, for we were all in Adam when he sinned, thus we all sinned. We die, because we are in Adam, sinners. Even before the law was given, people died.
Romans 5:12 Therefore, just as sin entered the world through one man, and death through sin, and in this way death came to all people, because all sinned—
13 To be sure, sin was in the world before the law was given, but sin is not charged against anyone’s account where there is no law. 14Nevertheless, death reigned from the time of Adam to the time of Moses, even over those who did not sin by breaking a command, as did Adam, who is a pattern of the one to come.
That said, however, the condition of death of believer's children may be different. 2 Samuel 12:23, Acts 16:31. Many suppose that underage children are covered by a covenant relationship of the Parent with God (aka Bar Mitzvah), others suppose ALL children are covered by a special dispensation of God. Problem is, we don't know for sure. I hold the first, but reject the second, on logical grounds and scriptural understanding, but one can't be dogmatically sure either way.