It seems that there were two guy named Dositheus, of which only one is the Samaritan heresiarch.
About the latter, so Krauss writes:
Quelques Pères de l'Eglise parlent d'un Dosithée qui aurait vécu peu après l'apparition de Jésus et qui fonda une secte encore avant Simon le Magicien, l'arci-hérétique. Le pseudo-Clément rapporte, au nom de Nicétes, que Dosithée aurait créé une secte après le meurtre de Jean-Baptiste. Dosithée fut ensuite relégué dans l'ombre par Simon le Magicien, qui, dans l'histoire de la primitive église, parvint à une triste célébrité. L'historien de l'ancienne église qui mérite le plus de créance, Hégésippe, cité par Eusèbe, place Dosithée à la même époque; seulement, chez lui, le rapport entre Simon et Dosithèe est renverse, car il fait apparaître Dosithée après Simon, ce que Hégésippe, ainsi que certains savants l'ont remarqué, n'a pas fait intentionnellement.
C'est le moment de citer les données des sources samaritaines sur les Dosithéens. Aboul-Fath raconte d'abord — à sa façon — l'histoire de Simon le Magicien, qui se serait allié avec Philon d'Alexandrie contre les disciples de Jésus; immédiatement après, il fait surgir les différentes sectes de Dusis, qui, par conséquent, aurait vécu à l'époque des Apotres. Or, dans une autre chronique samaritaine, le fondateur de la secte s'appelle Dosthis ..., en sorte qu'il n'y a pas de doute que Dusis désigne également Dosithée. Nous devons, comme chez Hégésippe, corriger l'ordre chronologique, de manière à placer Dosithée avant Simon. Signalons aussi Origène, toujouts bien informé, qui compare l'apparition de Dosithée avec celle de Juda le Galiléen. De la sorte, on appelle notre attention sur le caractère messianique de Dosithée, si bien que Dosithée se présenta comme Messie chez ls Samaritains, à l'instar de Jésus chez les Juifs. Il est naturel qu'Origène ait préféré comparer le Samaritan avec Juda plutôt qu'avec Jèsus, la messianité de Jésus étant à ses yeux infiniment supérieure à toutes les apparitions de meme ordre.
A ce mouvement messianique chez les Samaritains s'adapt fort bien un récit de Josèphe (Antiq ., XVIII, 4, 1), d'après le quel il y eut vers 35 une émeute sanglante à Samarie, que le procurateur Ponce-Pilate réprima avec une rigueur non moins sanglante. A la suite de ces événements, Pilate fut rappelé. Malheureusement, Josèphe ne nous indique pas le nom du fauteur de cette émeute; cependant, à en juger par toutes les sources citées plus haut, ce devait être Dosithée.
So,
de facto, the more strong argument to think that the Samaritan false prophet slain by Pilate was precisely Dositheus is this quote by Origen:
For this reason, they loved him [==John the Baptist] quite justly, but they did not keep their love within bounds; for they kept wondering ''whether perhaps he was the Christ''. The apostle Paul warns against inordinate and irrational love when he says of himself, ''I fear that someone might have an opinion of me above what he sees or hears from em,a nd that the greatness of the revelations might exalt me'', and so on. Paul feared that even he might fall into this error. So he was unwilling to state everything about himself that he knew. He wanted no one to think more of him than he saw or, going beyond the limits of honor, to say what had been said about John, that ''he was the Christ''. Some people said this even about Dositheus, the heresiarch of the Samaritans; others said it also about Judas the Galilean. Finally, some people burst forth into such great audacity of love that they invented new and unheard of exaggerations about Paul.
For, some say this, that the passage in Scripture that speaks of ''sitting at the Savior's right and left'' applies to Paul and Marcion: Paul sits at his right hand and Marcion at his left.
(
Origen, Hom. in Luc., 25)
Reading these words by Origen about so many rival Christs, one may remember the words of Robert Price about many rival Christs being at the origin of Christianity, merging at the end into only one fictional Jesus. Surely Origen is evidence of that.
Origen mentioned Dositheus also here, in
Contra Celsum 1:57:
And after him, in the days of the census, when Jesus appears to have been born, one Judas, a Galilean, gathered around him many of the Jewish people, saying he was a wise man, and a teacher of certain new doctrines. And when he also had paid the penalty of his rebellion, his doctrine was overturned, having taken hold of very few persons indeed, and these of the very humblest condition. And after the times of Jesus, Dositheus the Samaritan also wished to persuade the Samaritans that he was the Christ predicted by Moses; and he appears to have gained over some to his views.
Note that in both the passages Origen would like to compare Dositheus not with Jesus, but with Judas the Galilean. It is very suspicious the emphasis on
"And after the times of Jesus", as if Origen wanted to distance Jesus from Dositheus even from a chronological POV.
So Pilate was connected with Dositheus, in a first time.
Later Dositheus became the rival of the Christian Jesus
in a time when the anti-demiurgists coloured Dositheus with anti-demiurgist tints, since the Clementine Homilies, book II, make it clear that Dositheus contended with Simon Magus the anti-demiurgist title of 'Standing one'.
That this title was interpreted as anti-demiurgist, we have evidence in this passage:
And sometimes intimating that he [=Simon Magus] is Christ, he styles himself the Standing One. And this epithet he employs, as intimating that he shall always stand, and as not having any cause of corruption so that his body should fall. And he neither says that the God who created the world is the Supreme, nor does he believe that the dead will be raised. He rejects Jerusalem, and substitutes Mount Gerizzim for it. Instead of our Christ, he proclaims himself.
(2:22)
https://www.newadvent.org/fathers/080802.htm
So the name of Pilate passed from the Samaritan tradition about Dositheus to the Christian tradition behind the our Gospels, as effect of the growing rivalry between Christians and anti-demiurgists.
It was necessary to point out that Pilate crucified the davidic Jesus called Christ, not the Samaritan imposter Dositheus.