There was no entity known as the "Catholic Church" for hundreds of years.
This is like citing a letter by George Washington referring to the "American army," saying it used "American" as an adjective, not "American army" as a noun, and then concluding that there was no entity known as the "American army" for hundreds of years.
The argument is specious and silly.
What event happened "hundreds of years" later which caused the Catholic Church suddenly to pop into being (which event went completely unnoticed by contemporaries, BTW) ... ??
Whatever it was, it had happened by the time Augustine came around, because he wrote:
"And last, the very name Catholic, which, not without reason, belongs to this Church alone, in the face of so many heretics, so much so that, although all heretics want to be called ‘Catholic,’ when a stranger inquires where the Catholic Church meets, none of the heretics would dare to point out his own basilica or house" St. Augustine (Against the Letter of Mani Called "The Foundation" 4:5 [A.D. 397]).
And it certainly had happened by the time St. Cyril of Jerusalem had happened on the scene, because he wrote:
And if ever thou art sojourning in cities, inquire not simply where the Lord's House is (for the other sects of the profane also attempt to call their own dens houses of the Lord), nor merely where the Church is, but where is the Catholic Church. For this is the peculiar name of this Holy Church, the mother of us all, which is the spouse of our Lord Jesus Christ, the Only-begotten Son of God ... (St. Cyril of Jerusalem, lecture XVIII)
St. Cyril died in 386.
And it must have happened by the time Tertullian was around, because he wrote:
"Where was [the heretic] Marcion, that shipmaster of Pontus, the zealous student of Stoicism? Where was Valentinus, the disciple of Platonism? For it is evident that those men lived not so long ago—in the reign of Antonius for the most part—and that they at first were believers in the doctrine of the Catholic Church, in the church of Rome under the episcopate of the blessed Eleutherius, until on account of their ever restless curiosity, with which they even infected the brethren, they were more than once expelled" (Demurrer Against the Heretics 30 [A.D. 200]).
And with that, we're well before Constantine, so I rest my case.
Bump and thanks!
It is well that you rest your case.