I'm surprised to hear that schools explicitly state that they require master's degree. To my knowledge a master's degree is not a requirement for most pubpol PhD's. People entering Pubpol PhD programs tend to have a few years of experience and some do have MPP's. However, it is not a requirement and I'm sure you can sell your research experience at the UN.
I would say finding a Pubpol program with a strong focus on international development is the harder task since most pubpol programs focus more on domestic (social, health, urban) policies. If you want to focus on international policy issues I would suggest that you look into the affiliated faculties in poli sci or econ and check if they research on international development issues.
KSG is obviously the best choice. WWS would be good as well. I don't think Chicago's Harris School focuses on international development isues. Michigan's Ford School would be OK too. Maryland has international stuff too, but I believe it's focused more on security issues. LBJ(Texas Austin) focuses more on social policy. Duke has a new program so I can't tell what it's going to be like but presently it does not have much faculty focused on international development. NYU's Wagner School is mostly social and urban. CMU's Heinz as well. I'm not sure of Syracuse's Maxwell School. RAND's program is strong on security and health but not much development issues.
If you really are into development and are certain you don't want to do economics than you pool becomes much smaller.
I haven't done this but you could look for Development Think Tanks and look where the people did there PhDs.