Technically it is the Latin equivalent of the Hebrew Adonai and the Greek Kyrios, and which is not the word for God, but can be used for both civil rulers and as denoting the functional title for God who is the Divine Master. Elizabeth did not expressly call Jesus her God, but while Mary functionally was the mother of Israel - "according to the flesh" - a distinction the Holy Spirit is careful to make in Rm. 9:5 when speaking of Christ coming out of Israel, yet as a uncritically used formal title it goes beyond the langiage of Scripture as does Co-redemptrix. Maybe you like the latter also.
*"Dominus, plural Domini, in ancient Rome, “master,” or “owner,” particularly of slaves. The name became the official title for the emperor, beginning with Diocletian, who reigned from ad 284 to 305; and thus he and his successors are often referred to as the dominate (dominatus), as contrasted with the earlier principate (principatus) of Augustus and his successors. Some earlier emperors, such as Caligula (reigned ad 37–41), however, also had used the title. By Trajan’s day it was the common form of address to the emperor. - https://www.britannica.com/topic/dominus