There is a strong evidence that the meaning of monogenes (only one) was turned into monogenetos (only-begotten) by the mainstream Church slowly. Both words have different meanings, and I'm not sure if the early church writers used monogenitus, but they did change it to unigenitus in Latin. Jarome's Vulgate translation played a vital role in this change of official theology.
According to one authority on New Testament Greek “monogenos is literally ‘one of a kind,’ ‘only,’ ‘unique’ (unicus), not ‘only-begotten,’ which would be monogennetos (unigenitus).”(J Moulton, G. Milligan, Vocabulary of the Greek Testament. pp. 416-417)
This conception of begotten must have started even in the first century pagan or Gentile Churches of heresies. However, we can find the oldest evidence from Justin Martyr of the 2nd century. It seems the mainstream or ruling sect of Christians could never accept the eternal divine coequal nature of Christ, thus they had to turn him into a begotten, formed, temporal creation, like their pagan mythological gods. This is quite the same as the Arian belief that Christ was made. It doesn't make much difference with begotten, since begetting too is a form of bringing forth into existence, though not with the help of raw material as the organisms being created from dust. The only difference between Arius and others was that Arius argued that the Son was created ex nihilo, out of nothing, whereas the leading authorities argued he was created from the substance of the father, i.e. begotten like a creature is begotten with the same substance of the species.
- Dialogue of Justin, Philosopher and Martyr, with Trypho — Justin Martyr
"I shall give you another testimony, my friends," said I, "from the Scriptures, that God begat before all creatures a Beginning,  [who was] a certain rational power [proceeding] from Himself, who is called by the Holy Spirit, now the Glory of the Lord, now the Son, again Wisdom, again an Angel, then God, and then Lord and Logos; and on another occasion He calls Himself Captain, when He appeared in human form to Joshua the son of Nave (Nun). For He can be called by all those names, since He ministers to the Father's will, and since He was begotten of the Father by an act of will;  just as we see  happening among ourselves: for when we give out some word, we beget the word; yet not by abscission, so as to lessen the word  [which remains] in us, when we give it out: and just as we see also happening in the case of a fire, which is not lessened when it has kindled [another], but remains the same; and that which has been kindled by it likewise appears to exist by itself, not diminishing that from which it was kindled. The Word of Wisdom, who is Himself this God begotten of the Father of all things, and Word, and Wisdom, and Power, and the Glory of the Begetter, will bear evidence to me, when He speaks by Solomon the following: If I shall declare to you what happens daily, I shall call to mind events from everlasting, and review them. The Lord made me the beginning of His ways for His works. From everlasting He established me in the beginning, before He had made the earth, and before He had made the deeps, before the springs of the waters had issued forth, before the mountains had been established. Before all the hills He begets me. God made the country, and the desert, and the highest inhabited places under the sky.
Not all Church fathers believed in the begotten doctrine. Ignatius wrote in A.D. 110 that Jesus was gennetos kai agennetos meaning “begotten and not begotten.” (Revised Bauer-Arndt-Gingrich Greek Lexicon p. 156) Ignatius meant that in reference to the incarnation Jesus was “born of a woman” (Galatians 4:4) or “begotten” but in reference to his eternal status he was “not begotten.” It is clear that Ignatius believed according to the Gospels that Christ was begotten only in his flesh, in the incarnation, he was not-begotten in his deity.
It seems some Christian leaders were using begotten for the Son, as the begotten God to contrast him with the unbegotten Father. Only the Father God of unbegotten, uncaused cause. According to the early Christians, Christ was begotten (created or made into existence) before the creation of the world, like a fire is kindled. It seems they could never accept the eternal self existence of Christ or the co-equality in the Trinity, and the divine indivisibility or aseity, hence the Arian and subordination believes persist even to this day among all the Christian sects.
Pagan influential leaders were converting to the Church and were made Bishops and architectures of the official Creeds of the Church, like Augustin and Hilary of Poitiers in the fourth century. Hilary of Poitiers (ca. 315-367 CE) in his De Trinitate, also uses the term unigenitus for Christ.
The scripture reference of monogenes was changed by Jerome officially into begotten, in his new Vulgate translation. The Vetus Latina, Old Vulgate translation contained unicus (only) not unigenitus, that Jerome used in 4th century.
KJV alternate rendering of the Greek monogenes ( John 1:14 ,John 1:14,1:18; John 3:16 ,John 3:16,3:18; Hebrews 11:17; 1 John 4:9 ). Elsewhere the KJV rendered the term “only” (Luke 7:12; Luke 8:42; Luke 9:38 ). The phrase “only begotten” derives directly from Jerome (340?-420 A.D.) who replaced unicus (only), the reading of the Old Latin, with unigenitus (only begotten) as he translated the Latin Vulgate. Jerome's concern was to refute the Arian doctrine that claimed the Son was not begotten but made. This led Jerome to impose the terminology of the Nicene creed (325 A.D.) onto the New Testament.
(Holeman Bible dictionary, 1991)
When we have to do with living beings—men or animals—the meaning ‘born,’ ‘begotten’ is, of course, congruous, but there is no emphasis whatever attached to this side. When Christ is designated μονογενὴς υἱός, the emphasis is laid not on the fact that He as Son was ‘born’ or ‘begotten’ (in contrast to being ‘created’ or ‘made’), but that He is the ‘only’ Son, that as Son of God He has no equal. The Latin translators were quite right when originally they rendered the expression υἱὸς μονογενής simply by filius unicus, not by filius unigenitus. It was the dogmatic disputes as to the inner essential relations between Christ and God, especially those raised by Arius, which first gave occasion for emphasizing the point that Christ as the Son of God was a ‘begotten’ Son, i.e. that He did not form part of the creation. After that it became a general custom to render μονογενής by unigenitus, ‘only begotten.’ In the original form of the so-called ‘Apostolic Symbol’—the ‘Old Roman Symbol’—we read: καὶ εἰς Χριστὸν Ἰησοῦν τὸν υἱὸν αὐτοῦ τὸν μονογενῆ τὸν κύριον ἡμῶν; and in the Latin text, which in all probability belongs to the same date (i.e. in any case some time in the 2nd cent.): ‘et in Christum Jesum filium eius unicum dominum nostrum.’
(Ferdinand Kattenbusch, Hastings, James. Entry for 'Only Begotten'. Hastings' Dictionary of the New Testament)
Fortunately they did not pervert the Greek since their official language was Latin, however it is a well known fact that various textual corruption in Greek manuscripts are caused by scribal tendency to conform to the Vulgate translation. I see these facts about the changing of unicus to unigenitus are undisputed from the Roman apologists, you can confirm with the Vulgate comparison. The English Bibles also conformed to the Latin, as a result various corrections took too long to take place in the critical English versions, and even today despite the evidence, some scholars refuse to accept the original meaning of the Greek word, due to the traditional perception about it.