It depends. I am making a similar decision right now (HKS MPP with no aid vs Columbia SIPA with $20K a year in aid). I want to go into consulting straight after school because I don't really feel comfortable working in foreign policy in a conservative government or a center left government (I'm VERY progressive haha) so I need to focus on paying down my debt instead. HKS seems to have better private sector/consulting placement than most other public affairs/IR schools. That is what is swaying me but the extra debt burden I would have going there scares me.
Since you want to work in the public sector anyways, there would not be much of an edge for HKS over SAIS. SAIS is in Washington which has an added advantage. So in all, it's up to you what you want to do immediately after grad school. SAIS may be better for public sector work. The choice is yours haha.