Does that mean that the Carmelites celebrate the Most Pure Heart of Mary on the 22nd instead of after the Sacred Heart?
That is my understanding. In the typical edition of the Carmelite Breviary published in 1938(?), the Office for the Most Pure Heart of the Blessed Virgin Mary was still in its old place, but in the hand Missal cited above the Mass for this Feast takes the place of the Mass
Audeamus instituted by Pope Pius XII for today's Feast Day. It is my guess that they must have also done the same thing for the Office.