Hello Friar David,
I do not believe the regular praying of at least a part of the Liturgy of the Hours is a requirement. The doctrinal section of The Rite of Blessing of and Enrollment in the Scapular of the Blessed Virgin Mary of Mount Carmel" as issued and confirmed by the Congregation for Divine Worship and for the Discipline of the Sacraments, November 29, 1996, says this:
Thus, becoming part of the fraternity of the Order of the Brothers of the Blessed Virgin Mary of Mount Carmel and receiving the scapular as a sign of such membership, is for the faithful an instrument of Christian life with special characteristics:
(c) To imitate the “praying Virgin” who “kept the word of God in her heart” (cf. Lk 2:19,51), setting aside some time to meet God in prayer, meditating on the mysteries of salvation, taking part in faith in the liturgy of the Church, especially the Eucharist, **reciting every day some liturgical hour or some psalms or the holy rosary. ** To see to it that living in the presence of the living God, every aspect of life becomes prayer and that prayer becomes life, at the same time remembering the obligations and work of their state of life.
And again here:
Appendix: Practical Norms for the Scapular Confraternity
Members are bound to set aside regularly time to be with God in prayer, frequent participation in the Eucharist, daily recitation of one of the hours of the liturgy or of some psalms or the rosary or other equivalent prayers.