I only now realized that there was a question or two for me here.
I am not a theologion or a bishop but I’ll attempt to answer some of the questions. There is always a risk that I may inadvertantly misrepresent the church’s position, I take full reponsibility for any errors. I ask your understanding and forgiveness in advance.
Byzantine Spirituality is not based upon the Augustinian tradition. Augustines influence was most directly on the West, and a great deal of the doctrinal formulations in the west are based upon Augustinian philosophy/theology. Ditto for Thomist Philosophy, (and de Molina) which is peculiar to the Western church.
Eastern Catholic Priests and Deacons are generally familiar with Augustinian Theology and Thomist Philosophy, it is usually part of their seminary training, as far as I know.
The Byzantine liturgy and prayers reflect the theology of the Greek East. Byzantine Theology is a match to Orthodox theology in almost every way, (with the exception that we understand Western theological constructs as a valid for them).
To reiterate, Catholic churches recognize each others theologies as equally valid, but different, perspectives on the same underlying Truths, or Dogmas. All Catholics have a right to receive the Eucharist in other Catholic Sui Iuris churches without any obligation to change their own understanding of doctrine.
One can convert directly into the Byzantine Catholic church if one has come to an understanding of the Faith as a Byzantine. People may transfer to the Byzantine Catholic church from the Roman Church if their understanding of theology matches Byzantine teaching and they follow the Byzantine liturgical life and calendar through a Byzantine parish. The Eparch (bishop) will usually require a 1 to 3 year waiting period and a favorable report from the parish priest before going ahead with a canonical transfer. The request to transfer must come from the layperson, and is not suggested by the clergy.
There are many Roman Catholics who register with Byzantine Catholic parishes and follow the Byzantine liturgical life for many years and do not ultimately transfer into the Byzantine church, there is no pressure to do so.
So, for various reasons, there are worshippers at Byzantine parishes that have an Augustinian understanding of theology, there is nothing to forbid them from holding these views, but that theological perspective will not normally be taught at the parish or be reflected in the homiles.
I hope that this helps somewhat.
In Christ,
Michael