Check out Our Lady of the Atonement Catholic Church and its parish school, the Atonement Academy, located at 15415 Red Robin Road, off of Babcock and 1604 W:
www.atonementonline.com
OLOA is in full communion with the Holy Father and the Catholic Church, though we use forms of prayer and worship similar to Anglicans. It is a Roman Catholic Archdiocese of San Antonio parish. Not all Anglican Use parishes belong to the territorial (arch)diocese, however. Some are parishes of the Ordinariate of the Chair of St. Peter, which the Holy See erected in 2012. The ordinariate is equivalent to a diocese, and its head, an ordinary, may be (and currently is) a presbyter (simple priest, as opposed to a bishop). In addition to Mass, we have Evensong (Vespers) offered regularly.
Also, St. Anastasia the Great Martyr Byzantine Catholic Community offers Divine Liturgy on the 1st, 3rd, and 5th Sunday of each month, unless noted. It is a bit farther out, closer to the northeast side:
sabyzcath.org
This community is under the jurisdiction of the Ruthenian Byzantine Catholic Metropolitan Archeparchy of Pittsburgh, whose latest metropolitan was appointed by Pope Benedict XVI in 2012. An eparchy is the Eastern Catholic equivalent of a diocese. We are served by a Roman Catholic priest, a Dominican, who has faculties in the Ruthenian church. When Divine Liturgy is not offered at the Byzantine mission, many will attend the Maronite church. Some of us, namely myself, will attend OLOA. Still others attend the territorial Latin (Roman) Catholic parishes.
There is also a Traditional Latin Mass community in San Antonio, non-TLM, but located at an archdiocesan parish under the patronage of St. Pius X. I have not been to this community yet, so I will allow someone else to vouch for it.
Finally, St. George Maronite Catholic Church offers Divine Liturgy each weekend. Sung liturgies at 11 o’clock AM. Like the Byzantines, they also have their own bishop but are in full communion with the Holy Father, currently Pope Francis.
www.stgeorgesa.org
(I encourage you to visit the parish in person, as it looks like this website has not been updated recently.)