Both Ellegaard and Earl Doherty have pointed out that Ignatius insisted on a dating of the crucifixion "under Pilate" in opposition to Docetists denying the carnal reality of Jesus.
In particular, Doherty, following Solomon Reinach on this point (when Reinach was still a mythicist, i.e. before that he read the Robert Eisler's works), thinks that Ignatius arrived so far to attack early "mythicist" adversaries.
Now, Ellegaard doesn't explain why Ignatius mentioned precisely Pilate inter alia, even if Ellegaard was very brilliant in recognizing that Ignatius introduced Pilate even before the writing of the first gospel.
I think that I have found the true reason behind the Ignatius's choice of Pilate and only Pilate.
In the epistle of St. Ignatius Ad Trallianos (§ 11), Simon Magus is called "the first-born Son of the Devil". Accordingly, Ignatius knew Simon Magus as basic icon of docetism.
This is surprising, since it means that, being Simon a Samaritan, and being one who was considered as the originator of a try of Samaritan co-optation of the Jesus myth, along a line that dates back to the Samaritan Dositheus and even John the Baptist (of which the tomb was placed in Samaria, Sebaste) then Ignatius, by arguing against Docetists and Simon Magus, was replying probably against the famous claim of the Magus, reported by Irenaeus:
As Paul_the_Uncertain has reported in his blog, Simon Magus was someway an early "mythicist" insofar he denied that a historical Jesus was in his (of Simon) place when he, Simon, apparently suffered on the cross.
Now, we have independent confirmation of a Samaritan cult of Jesus as "Son of Joseph". Simon Magus was probably part of that cult. It can't be a coincidence that the only Samaritan insurrectionist reported by Josephus was crucified precisely by Pilate, and that this insurrectionist was posing someway as the new Moses and probably as the Samaritan Messiah Son of Joseph. I am not saying that Simon Magus was the name of the Samaritan false prophet slain by Pilate. What I am saying is that Simon Magus was part of that Samaritan tradition that saw the Samaritan false prophet slain by Pilate as a messianic figure.
Simon Magus and the docetists could have denied that he, as the Samaritan Messiah, suffered really under Pilate, and in the same time they could have insisted that Simon Magus was "Jesus".
As reaction, Ignatius would have said the exact contrary: Jesus was real, i.e. he wasn't Simon, and Jesus was crucified really by Pilate, so taking Pilate from the Samaritan source represented by the his opponents.
The reader has to persuade himself/herself that my case is virtually stronger once we realize that Ignatius mentioned Simon Magus.