Q. 1. Is it appropriate for the Catholic Church to administer the Sacrament of Baptism during the Holy Mass on a regular basis?
A. 1. Your question has two parts to it:
a. Is it appropriate for the Catholic Church to administer the Sacrament of Baptism during the Holy Mass?
b. Can such be done on a regular basis?
Canon Law # 856 states, "Although baptism can be celebrated on any day, it is nevertheless recommended that it be celebrated ordinarily on Sunday or, if possible, at the Easter Vigil."
The only reference to administering the Sacrament of Baptism during the Holy Mass is at the Easter Vigil. The reference to Sunday does not indicate that it can be done during the Holy Mass.
The attitude of many priests and layperson is that since there is no clear instruction as to if the Sacrament of Baptism should be within a Mass or outside of Mass, then both can be applied. This reasoning is a grave error.
The Catholic Church has always thought that no one, not even a priest, may add, remove, or change anything in the liturgy on his own authority. In other words, no one can add the Sacrament of Baptism during the celebration of the Holy Mass for the sake of a novelty. It is forbidden. Read the following:
"Sacrosanctum Concilium #22: (1) Regulation of the sacred liturgy depends solely on the authority of the Church, that is, on the Apostolic See, and, as laws may determine, on the bishop. (2) In virtue of power conceded by law, the regulation of the liturgy within certain defined limits belongs also to various kinds of bishops' conferences, legitimately established, with competence in given territories. (3) Therefore no other person, not even a priest, may add, remove, or change anything in the liturgy on his own authority."
To answer the second part of the question, since it is not appropriate to celebrate the Sacrament of Baptism during the Holy Mass, then it is not appropriate to do it on a regular basis.