I think it very unlikely.
Pilate did not demonstrate a terribly deep understanding of or appreciation for Judaism - though he knew enough to make them mad (e.g. the sign on the cross).
Allowing that the Son of Man is a reference to the Book of Daniel (widely but not unanimously held), I can't imagine Pilate had ever read the prophecy, much less appreciated it. If the Jews themselves at the time weren't 100% sure what Jesus meant when He called Himself "Son of Man", surely Pilate, who didn't even entirely grasp what the "Messiah" was, wouldn't have made the connection to a lesser-known Jewish prophecy.
Why do I suggest the Jews weren't 100% sure what Jesus meant when He called Himself "Son of Man"? If they had known, the Sanhedrin would have tried to kill Him a lot more a lot sooner. His claims to be the Messiah & the Son of God are relatively few and far between in the Gospels. He calls Himself the Son of Man all the time.
Pilate appears to be playing on the people's sympathy by making them feel sorry for a poor, suffering man. As Nigel J noted, Pilate is emphasizing Jesus' scourged humanity rather than His Divinity.