Finding a “traditional” third order of Carmel

Recent apologetics answers by Michelle Arnold – Catholic Answers
I think the question that should take precedence in your discernment process is whether or not you want to actually be a Carmelite. Many people are attracted to Carmelite spirituality, love Carmelite saints, and appreciate the contributions of the Carmelites to the Church, but they are not necessarily suited to be Carmelites (i.e., members of the order according to their state in life). This may sound like a paradox but being a lay member of a religious order such as the Carmelites is not like joining a club but is instead agreeing to take on a way of life; taking on that way of life does not mean doing so on one’s own but by aspiring to be a member of an approved Carmelite community.
In the Carmelites, those laypeople who are members of a Carmelite order make a promise before their Carmelite community and a delegate of the Carmelite order to live the evangelical counsels of poverty, chastity, and obedience, in a manner appropriate to their lay state. If, for example, a person cannot in conscience promise to strive for the obedience appropriate to a lay member of a religious order, then he should not seek membership. While anyone can wear the small brown scapular that is a popular sacramental of the Church, the larger brown scapular you refer to is a sign of membership in the Carmelite order and non-members should not presume to wear it.
I say all this as prelude to your concern about finding a “traditional” third order of Carmel. To be a Carmelite, it is necessary for an aspirant to be willing to be taught and formed by fellow members of the order entrusted with that task. If a person is unteachable, according to the norms and charism of the religious order he aspires to, that is a sign that he is not called to that order. In other words, it is more important for a prospective Carmelite to be willing to be formed by the order as it exists than to find a particular group of Carmelites he likes.
That said, in my own local secular Carmelite community, I know of at least three members who regularly attend Mass at the parish in our diocese that exclusively offers the Tridentine Mass and the sacraments according to the traditional Roman rubrics. It is possible to be both a lay member of the Carmelites and a traditional Catholic.
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“If anyone comes to me, I want to lead them to Him.” –St. Edith Stein