I'm in a very similar situation to you - French & Japanese major in senior year of undergrad and considering grad programs in international development. I can't speak from experience, but I have a hunch that the university you got your undergrad degree from doesn't matter that much at all in masters admissions. I think it's as you said - your research experience in a position normally reserved for master's students, your coursework, and your expressed interest would count a lot more. I wouldn't worry too much about your university's reputation.
Are you applying for TAPiF? That can help a lot. If I remember correctly, there are even schools that give significant tuition discounts to participants of TAPiF (NYU I believe is one of them). Also, have you considered universities in France? It seems that Sciences Po, for example, has some great International Development master's programs.