Actually the most distinctive papal headgear, the one that popes used to be drawn wearing in political cartoons to distinguish them from ordinary bishops, is what is called the papal tiara, a tall conical shape with three crowns around it, one above the other. Of course the popes have rarely wore the papal tiara in the last fifty years or so, so younger people might not be familiar with it.

https://en.wikipedia.org/wiki/Papal_tiara

https://churchpop.com/2015/08/12/when-popes-wore-crowns-a-pictorial-history-of-the-papal-tiara/

The bishop's miters or mitres worn by Roman Catholic bishops (and by bishops of some Protestant denominations) mostly in religious ceremonies, were originally conical cloth hats, I believe. Over the centuries miters or mitres became taller and split into left and right parts and then the design was turned 90 degrees and they were worn split into front and back parts. The miters or mitres of eastern Orthodox bishops developed a different form.

https://en.wikipedia.org/wiki/Mitre

The wearing of mitres apparently began about the year 950, in Rome, far from Syria, and early mitres looked much less like fish hats of Dagon priests than modern mitres do.

From the seventeenth century much has been written concerning the length of time the mitre has been worn. According to one opinion its use extends back into the age of the Apostles; according to another, at least as far back as the eighth or ninth century while a further view holds that it did not appear until the beginning of the second millennium, but that before this there was an episcopal ornament for the head, in form like a wreath or crown. In opposition to these and similar opinions, which cannot all be discussed here, it is, however, to be held as certain that an episcopal ornament for the head in the shape of a fillet never existed in Western Europe, that the mitre was first used at Rome about the middle of the tenth century, and outside of Rome about the year 1000. Exhaustive proof for this is given in the work (mentioned in bibliography below), "Die liturgische Gewandung im Occident und Orient" (pp. 431-48), where all that has been brought forward to prove the high antiquity of the mitre is exhaustively discussed and refuted. The mitre is depicted for the first time in two miniatures of the beginning of the eleventh century; the one is in a baptismal register, the other in Exultet-roll of the cathedral at Bari, Italy. The first written mention of it is found in a Bull of Leo IX of the year 1049. In this the pope, who had formerly been Bishop of Toul, France, confirmed the primacy of the Church of Trier to Bishop Eberhard of Trier, his former metropolitan who had accompanied him to Rome. As a sign of this primacy, Leo granted Bishop Eberhard the Roman mitre, in order that he might use it according to the Roman custom in performing the offices of the church. By about 1100-50 the custom of wearing the mitre was general among bishops.

The first example of a mitre of the modern shape appears about 1150.

https://www.newadvent.org/cathen/10404a.htm

The worshp of Dagon, along with all other pagan religions, was discourageed and later forbidden by many decrees of different emperors in the 4th century and later centuries. Paganism became a minority religion practiced in private in about the 5th century BC in most parts of the Roman Empire. I would expect that the last fish hat was publicly worn by some priest of Dagon sometime in the 5th century.

So the present vague resemblance between western bishops' miters or mitres and the fish hats of Dagon priests is relatively new. It is certainly not the case that early Christian bishops were depicted wearing modern miters or mitres or anything else resembling fish hats like they would have been depicted if they were priests of Dagon.

I sort of suspect that the originator of the theory you mentioned was just making up a goofy story to see if anyone would believe it, though the persons who repeat it at the present may sincerely believe it.

Dagon was worshipped for a few millennia by ancient Middle eastern people, mostly in Syria. After the Roman Empire converted to Christanity in the 4th century all pagan reliigons including worship of Dagon were discouraged, and later prohibited, and later persecuted.

When Christianity was forming Dagon waorship was common in a relatively small region of the Roman empire, mostly parts of Syria, while worshp of the Graco-Roman gods was spread over all of the Roman Empire and was the official religion of the army and government and all the upper class members of society. There were also larges numbers of early Christians in the Parthan and later Sassanian Empire, where Zoroastrianism was the state religion and other religions, including Christianity, were sometimes persecuted.

So if the ancient Christians were going to imitate various aspects of non Christian reliigons, they would mostly copy Judiaism, which Christianity ws an offshoot of, and they would secondly copy the two high prestige religions in the lands where they lived, Graeco-Roman polytheism and Zoroastrian monotheism. A comparativley local cult like Dagon worship would probably be way down on their list of religions to copy from.