First Chern class of a flat line bundle

I noticed that someone voted this up today. Since this might indicate that someone else is interested in the answer, I thought I'd remark that Oscar Randal-Williams and I worked out a proof of this when I visited him earlier this year. A version of this proof can be found in Section 2.2 of my paper

The Picard group of the moduli space of curves with level structures, to appear in Duke Math. J.

which is available on my webpage here.

(marked community wiki since it feels weird to get reputation for answering my own question)

It's probably not exactly what you want (in particular, they're dealing with real bundles and the Stiefel Whitney classes), but something sort of close is discussed in the appendix to

MR2003827 (2004h:53116) Ho, Nan-Kuo(3-TRNT); Liu, Chiu-Chu Melissa(1-HRV) Connected components of the space of surface group representations. Int. Math. Res. Not. 2003, no. 44, 2359–2372. 53D30 (22F05 57N05)