BTW, MAPLE's limit command also tells me that the limit is $0$ which would be enough for my purpose, although I don't really trust MAPLE in this since I don't know how it computes these things.

There might be a tiny problem with this solution though: The Mordell integral as such is only defined for $\tau$ in the upper halfplane, in particular not in any neighbourhood of $0$. So a Taylor expansion around $0$ is not very well possible, is it? MAPLE probably doesn't care for that though...