In the hypothesis of Incarnation without sin, it is obvious that the Incarnated Word would not have been rejected, mistreated and killed by people. Not only the passion and death of the Incarnated Word would not have been necessary for the atonement of sins, but also it would not have been even possible, because people in the original state of holiness and justice would have recognized, worshipped and loved Him.
In this hypothetical case, the Word would still have assumed a human nature "for us men, and for our salvation" as the Nicene creed says, understanding salvation not in the negative way of delivering people from hell, but in the positive way of making people "partakers of the divine nature" (2 Peter 1:4), members of the Word and temples of the Holy Spirit, and thus able, when glorified at the end of their time on earth, to see God "as He is" (1 John 3:2), "face to face" (1 Cor 13:12).
To note, the case is **not** that the Incarnation of the Word is the **only** way in which God can make people partakers of the divine nature even if they did not sin, but that it is the most fitting way. Just as the case was not that, once people sinned, the loving obedience of the Incarnated Word to God the Father to the point of death on a cross was the only way in which God could forgive people for their sins and make them again partakers of the divine nature, but that it was the most fitting way (Summa Theologica, Part III, Question 46, Articles 1-3) [1].
The notion that the Word assuming a human nature is the most fitting way for God to make people partakers of the divine nature, even if people had not sinned, seems fairly self-evident. It is also evident in several texts from the Church Fathers which refer to the Incarnation as the divine course of action for our theosis without connecting it necessarily with the Passion of the Incarnated Word for the atonement of sins, with some of those texts referred in paragraph 460 of the Catechism [2]. An ancient antiphony of Latin liturgy praises this divine course of action, again without making a necessary connection to the Passion for the atonement of sins, by calling it "wonderful exchange" ("admirabile commercium") [3]. For further reference, there is ample current literature on the subject, from brief [4] to extensive [5].
In this hypothetical scenario, the Incarnated Word would still have given us an example of radical Self-giving to God the Father, even if that Self-giving would not have involved an obedience to the Father to the point of death. Clearly for the Incarnated Word to give Himself to God the Father in loving obedience, and out of that love, to the people whom the Father loves, it is not intrinsically necessary that the Father commands the Word to let people kill Him! Though suffering to the point of death in loving obedience to God is the highest way in which radical self-giving to God may be realized, clearly it is not the only way.
**References**
[1] http://www.newadvent.org/summa/4046.htm
[2] http://www.vatican.va/archive/ENG0015/__P1J.HTM
[3] First antiphon of Vespers and Lauds of the pre-1969 feast of the Octave of the Nativity of the Lord / Circumcision of the Lord, and first antiphon of Vespers of the current feast of St. Mary Mother of God, both feasts on January 1st: "O wonderful exchange: The Creator of the human race, assuming an ensouled body, deigned to be born of a Virgin, and coming forth as a man without intervention of [male] seed, has granted to us his Divinity." ("O admirabile commercium: Creator generis humani, animatum corpus sumens, de Virgine nasci dignatus est, et procedens homo sine semine, largitus est nobis suam deitatem.")
[4] E.g. https://www.hprweb.com/2010/12/o-admirabile-commercium-the-true-christmas-exchange/
[5] E.g. https://www.ewtn.com/catholicism/library/o-marvelous-exchange-12607