What will be the role of (non-baptized) cafeteria Catholics in the Catholic Synodal Way?
In one way, this is a non-question, if you take into consideration of the non-baptized. Enterance into the Church starts with baptism. If someone is still unbaptized, they are thus not considered members of the Catholic Church.
1213 Holy Baptism is the basis of the whole Christian life, the gateway to life in the Spirit (vitae spiritualis ianua), and the door which gives access to the other sacraments. Through Baptism we are freed from sin and reborn as sons of God; we become members of Christ, are incorporated into the Church and made sharers in her mission: "Baptism is the sacrament of regeneration through water in the word." - Catechism of the Catholic Church
In this light, the objective of this diocesan phase is to consult the People of God so that the Synodal Process is carried out through listening to all the baptized. By convoking this Synod, Pope Francis is inviting all the baptised to participate in this Synodal Process that begins at the diocesan level. Dioceses are called to keep in mind that the main subjects of this synodal experience are all the baptised. Special care should be taken to involve those persons who may risk being excluded: women, the handicapped, refugees, migrants, the elderly, people who live in poverty, Catholics who rarely or never practice their faith, etc. Creative means should also be found in order to involve children and youth. - Vademecum for the Synod on Synodality
My thoughts about the so-called cafeteria Catholics is that there will be some of their numbers registered for the actual synod itself.
The term cafeteria Catholic is applied to those who assert a Catholic faith yet dissent from one or more doctrinal or moral teachings of the Catholic Church or who are viewed as dissenting by those using the term. Cafeteria Catholics "...pick and choose what and how they will believe-as they do in cafeteria lines." and what they choose to accept or reject can be any of the teachings within the Catholic church. Examples range from Catholics who choose not to follow one or more of the seven sacraments (Baptism, Confirmation, Eucharist, Penance, Anointing of the sick, Holy Orders, and Matrimony) (i.e., Some Cafeteria Catholics do not think confession to a priest is necessary to have sins forgiven) to choosing not to follow other official positions currently upheld by the Catholic Church. Other examples include Catholics who are accused of dissenting from any or all of the Church teachings on human sexuality and things related (the so-called "pelvic issues", i.e., what it has to say about abortion, birth control, divorce, premarital sex, masturbation, pornography, prostitution and the moral status of homosexual acts) or, alternately, those who demonstrate no concern for any moral issues except abortion and contraception.
Use of the term
The term is most often used by conservative Catholics critical of progressive Catholics. The term has been in use since the issuance of Humanae Vitae, an official document that propounded the Church's opposition to the use of artificial birth control and advocates natural family planning.
It is often a synonymous phrase for "Catholic-in-name-only" (or CINO), "dissident Catholic", "heretical Catholic", "cultural Catholic"/"cultural Christian", "à la carte Catholic", or "liberal Catholic".
The term has no status in official Catholic teachings. However, the practice of denying adherence to the sexual morality of the Church has been criticized by Pope John Paul II stated in his talk to the Bishops in Los Angeles in 1987:
It is sometimes reported that a large number of Catholics today do not adhere to the teaching of the Catholic Church on a number of questions, notably sexual and conjugal morality, divorce and remarriage. Some are reported as not accepting the clear position on abortion. It has to be noted that there is a tendency on the part of some Catholics to be selective in their adherence to the Church's moral teaching. It is sometimes claimed that dissent from the Magisterium is totally compatible with being a "good Catholic," and poses no obstacle to the reception of the Sacraments. This is a grave error that challenges the teaching of the Bishops in the United States and elsewhere.
During a morning Mass at the Domus Sanctae Marthae, Pope Francis, speaking rather of half-hearted Catholics, said, "They may call themselves Catholic, but they have one foot out the door."
Some bishops themselves are considered as cafeteria Catholics if the above definition is to be invoked. Thus cafeteria Catholics will be involved. Remember that Pope Francis said that ”special care should be taken to involve those persons who may risk being excluded: women, the handicapped, refugees, migrants, the elderly, people who live in poverty, Catholics who rarely or never practice their faith, etc.”