Philo says about Pilate only this 'Gospel' clue:
...his continual murders of people untried and uncondemned...
I am not sure the translation is correct, because my Italian translation reads something as:
...his frequent executions without trial...
But is it sufficient to make a case for 'Mark' reading that passage and saying:
"Eureka! Pilate is the best name for the Roman killer of Jesus!"
Another possibility is that the entire Philonian passage refers the same Josephian episode about Pilate and the imperial shields. I underline in particular the point of interest who is not found in the parallel episode in Josephus:
XXXVIII. (299) "Moreover, I have it in my power to relate one act of ambition on his part, though I suffered an infinite number of evils when he was alive; but nevertheless the truth is considered dear, and much to be honoured by you. Pilate was one of the emperor's lieutenants, having been appointed governor of Judaea. He, not more with the object of doing honour to Tiberius than with that of vexing the multitude, dedicated some gilt shields in the palace of Herod, in the holy city; which had no form nor any other forbidden thing represented on them except some necessary inscription, which mentioned these two facts, the name of the person who had placed them there, and the person in whose honour they were so placed there. (300) But when the multitude heard what had been done, and when the circumstance became notorious, then the people, putting forward the four sons of the king, who were in no respect inferior to the kings themselves, in fortune or in rank, and his other descendants, and those magistrates who were among them at the time, entreated him to alter and to rectify the innovation which he had committed in respect of the shields; and not to make any alteration in their national customs, which had hitherto been preserved without any interruption, without being in the least degree changed by any king of emperor. (301) "But when he steadfastly refused this petition (for he was a man of a very inflexible disposition, and very merciless as well as very obstinate), they cried out: 'Do not cause a sedition; do not make war upon us; do not destroy the peace which exists. The honour of the emperor is not identical with dishonour to the ancient laws; let it not be to you a pretence for heaping insult on our nation. Tiberius is not desirous that any of our laws or customs shall be destroyed. And if you yourself say that he is, show us either some command from him, or some letter, or something of the kind, that we, who have been sent to you as ambassadors, may cease to trouble you, and may address our supplications to your master.' (302) "But this last sentence exasperated him in the greatest possible degree, as he feared least they might in reality go on an embassy to the emperor, and might impeach him with respect to other particulars of his government, in respect of his corruption, and his acts of insolence, and his rapine, and his habit of insulting people, and his cruelty, and his continual murders of people untried and uncondemned, and his never ending, and gratuitous, and most grievous inhumanity. (303) Therefore, being exceedingly angry, and being at all times a man of most ferocious passions, he was in great perplexity, neither venturing to take down what he had once set up, nor wishing to do any thing which could be acceptable to his subjects, and at the same time being sufficiently acquainted with the firmness of Tiberius on these points. And those who were in power in our nation, seeing this, and perceiving that he was inclined to change his mind as to what he had done, but that he was not willing to be thought to do so, wrote a most supplicatory letter to Tiberius. (304) And he, when he had read it, what did he say of Pilate, and what threats did he utter against him! But it is beside our purpose at present to relate to you how very angry he was, although he was not very liable to sudden anger; since the facts speak for themselves; (305) for immediately, without putting any thing off till the next day, he wrote a letter, reproaching and reviling him in the most bitter manner for his act of unprecedented audacity and wickedness, and commanding him immediately to take down the shields and to convey them away from the metropolis of Judaea to Caesarea, on the sea which had been named Caesarea Augusta, after his grandfather, in order that they might be set up in the temple of Augustus. And accordingly, they were set up in that edifice. And in this way he provided for two matters: both for the honour due to the emperor, and for the preservation of the ancient customs of the city.
I note in particular that:
[the shields] mentioned these two facts, the name of the person who had placed them there, and the person in whose honour they [the shields] were so placed there
'Mark' (author) could have thought:
'the name is Pilate and the person in whose honour Pilate had disturbed the Jews was... ...Jesus called King of Jews!"
Hence, the point of the thread remains: the shields episode was decisive to persuade 'Mark' to derive midrashically the Roman trial of Jesus from it.