How is the Hebrew month of Tishri/Tishrei pronounced?

Pointed medieval manuscripts (e.g. of the Mishnah) all read Tishri. Classical and medieval liturgical rhymes all assume an ending of -ri. Latin transliterations of the name also end in -ri (e.g. in the Liber Erarum: see P. Nothaft, Medieval Latin Christian Texts on the Jewish Calendar, 2014, p. 91 – I have a lot more references on all this).

The pronounciation Tishrei is a later development (Ashkenazi, I have been told). Does anyone know when it is first documented?