This rather complex problem was first exposed by Nicolaus von Kues, Lorenzo Valla and the English bishop Reginald Pecock (in his writing against the Lollards). The whole background story of the donation is dependent on the Acts of Sylvester, which narrates the baptism of Constantine by Sylvester after the victory at the Milvian bridge (cf. Wilhelm Pohlkamp, Textfassung, Literarische Formen und geschichtliche Funktion der römischen Sylvesterakten, in Francia 19/1 (1992), pp.115-196; digitally accessible in the Bavarian State Library). The validity of these actus Sylvestri is in turn assured by the pseudo-Gelasian decretum, a Gallic index of permissible books from the 6th century (cf Erich von Dobschütz, Decretum Gelasianum, in the series Texte und Untersuchungen zur Altchristlichen Literatur, Bd xxxviim 3m Leipzig 1911, digitally accessible at the Internet Archive) that eventually made its way into the Gratian judicial corpus.
The core problem was that Eusebius of Cesarea, though later condemned as Arian, had himself been the Church historian of choice of Constantin, as Reginald Pecock pointed out, and had narrated no baptism of Constantine in Rome (neither after the battle nor due to leprosy), but placed that even in Nicomedia toward the end of the Emperor's life.