NOPE.
Where did Mark obtain the name, because Mark wrote first, as we know from reading Justin Martyr, who cites, without reference, several passages that correspond most closely to Matthew, the writer who in turn, used Mark as a source.
"Paul" doesn't appear on the scene until the late second century, when his epistles are referenced by the Bishop of Lugdunum, who conveniently had traveled to Rome, on that very day, when the Roman army captured Irenaeus' predecessor, Pothinus, and executed him, for having committed the crime of practicing Christianity. Was Irenaeus not guilty of the same practices as Pothinus?
Actually that is not entirely true as we have other references to Paul before 180 CE:
"By reason of jealousy and strife Paul by his example pointed..."
"Take up the epistle of the blessed Paul the Apostle." - First Clement (c80-c140)
The seven writings of Ignatius of Antioch (d 98 to 117 CE) contain references to Paul
"In his Dialogue with Trypho, Justin extensively quotes the Jewish scriptures and includes several citations of logia of Jesus. Furthermore, while explicit citations from Paul are peculiarly absent from the text, Justin, writing from Rome, certainly knows Paul's writings in detail and uses them. Indeed, it seems that the Dialogue provides a perfect occasion for him to employ Paul because in it he addresses the relationship between Judaism and the church, a central topic in both Romans and Galatians. Besides the appearance of Pauline quotations, several of Justin's arguments directly rely on Paul's thinking. For example, Justin probably has Galatians 3 before him as he composes Dialogue 95–96. Oskar Skarsaune's analysis of Justin's writing also indicates that Romans is one of Justin's preferred sources for quotations of the Jewish scriptures; that is, he sometimes quotes the Jewish scriptures as they appear in Paul rather the LXX. He draws especially from the Jewish scriptures quoted in Romans 2–4 and 9–11 because the chapters examine the problem of Torah and the Jews' rejection of the gospel, also two important issues in the Dialogue." - Rodney Werline (1999). The Transformation of Pauline Arguments in Justin Martyr's Dialogue with Trypho. Harvard Theological Review, 92, pp 79-93. doi:10.1017/S0017816000017867.
Marcion of Sinope's heretical bible of c140 supposedly used Paul's writings and credited what came to be known as Luke to Paul.
So we have possible references to Paul way before the "late second century" date presented.