You could google your way to a lot of facts about the very many different days on which this Feast has been kept, as 'Gallican' bishops, and, later, Popes, granted it to various places or orders. The earliest date appears to be October 20; a date which enjoyed the favour of S John Eudes as early as 1647. But there seems to have been an increasing tendency to fix it on the Octave Day of the Assumption; or the Sunday within ... or the Sunday after ... that Octave. My 1874 Breviary has it then (among, of course, the observances pro aliquibus locis).
But other old liturgical books, as late as 1957, assign it to the Saturday after the Feast of the Sacred Heart. It was, I believe, on this day in the old Carmelite Rite.
In 1942, "gravissimas miseratus aerumnas quibus christiani populi ob ingruens immane bellum affliguntur" Pius XII consecrated the Human Race to the Immaculate Heart, and ordered its feast to be kept in the Universal Church on August 22, discarding the old Octave Mass of the Assumption. It will be seen that this represented, in calendar terms, a reversion to one nineteenth century date of the Feast of the Most Pure Heart. But the Bugnini idea of using the Saturday after the Feast of the Sacred Heart was, you will have noticed, not a totally new idea. It was a reversion to the date employed until the middle of the twentieth century for the feast in the days when our Lady's Heart was described as 'Most Pure' rather than as 'Immaculate' and was still pro aliquibus locis rather than universal.
If the EF and OF calendars are one day to be harmonised, it would seem to me appropriate to keep this lovely and Biblical feast on the day after the Sacred Heart, where it was until 1944 and then after 1970, not least because that would harmonise with the 'First Saturday' devotion. It would also emphasise that the Hearts of Jesus and Mary are two Hearts which, as we say, beat as one.
The Octave Day of the Assumption, and Maria Regina, could then fight it out for August 22. My instinct would be to call the day by its proper name, the Octave Day of the Assumption, but to incorporate some features of Pius XII's Office or of the Liturgia horarum, such as the hymn O quam glorifica, which dates from the ninth century and was originally proper to Assumption Day itself.
May 31 should revert to being the Feast of our Lady Mediatrix of All Graces, as it increasingly was in many places where so granted by indult, until Pius XII parked Maria Regina on that date. (When Bugnini foolishly transferred the Visitation to May 31, Dom Anselmo Lentini did make a deliberate attempt to keep a memory of this earlier celebration of our Lady of Grace by including, in a hymn he composed for the Visitation, the stanza Teque felicem populi per orbem/ semper, O Mater, recitant ovantes/ atque te credunt Domini favorum/ esse ministram.)