LaughingCat

Seconding @Nonprofitguy, but taking it broader: networking is important no matter where you want to go, not just if you want to get into some of the big charter school education networks. Go back onto LinkedIn, and see where the alumni of all the schools that you're interested in are currently clustered (cities or industries or even individual firms you're interested in). That LinkedIn search will help you visualize the network you're going to rely on when you graduate (or even while you're a student, for internships and such), so it's a good idea to understand what that looks like right now. 
Speaking of looking at where alumni are, @LaughingCat, your comments so far seem to reflect more a regional understanding of public policy schools, which is definitely grounded in truth and a pretty common belief, but far from the whole picture. Having graduated from Duke Sanford, I pretty much know only one person that's working in Atlanta, but obviously a boatload in NC government, the second largest contingent in DC, and many people scattered throughout New York, Baltimore, San Francisco, etc. The same goes for schools like Michigan Ford, because the alums don't just all flock to Chicago - the geographic distribution is all over, really, as you'll see from LinkedIn. What you may also find is the same organizations or industries popping up time and time again. For example, Deloitte's federal practice makes it a point to recruit heavily from Sanford because of the alumni network, to the point where I remember five Deloitte networking events throughout the year. 
Also, not to quibble over details, but often, it's a friend-of-a-friend who might prove the most useful resource to you, so even if there isn't a critical mass of education folks where you end up, you can still really use that network for your own ends. (Theory of weak ties, thanks core sociology.) Though some alumni I met had very different careers and interests from me, they often knew of where I could find someone/ some org that was a good match. That's how on one occasion, I found the org I interned with between my first and second years, and on another occasion, made friends with a much older alumna. You never know where networking might lead you, which in my view makes it well worth the effort of a little small talk. 

Well it's related to your goals of working in K-12 education. You'll quickly find none of the roles require you to have an MPP/MPA. I have many friends who work in the big charter networks in East coast (mostly through TFA: Achievement First, Brooke, KIPP, Northstar, Success, Uncommon, etc.) that have taken on senior leader positions. Some have gone on to work for Education Pioneers, TFA, TNTP, etc. Maybe 30% of them have graduate degrees and if they do, it happens to be a Masters in education, typically from Harvard or Teachers College. 
In case you think this is just purely anecdotal, just do quick LinkedIn searches of different charter networks you are thinking about and you'll see a similar picture. This is often because these networks often promote their star teachers internally, something like: classroom teacher => Leadership fellow => Vice Principal => Principal. Meaning if you actually want to get involved in this sector, you want to go to a grad school where there is a high concentration of people who are also going to work in this field in the area you want to work in, increasing the opportunity for getting different type of roles in different type of educational organizations.

I think Ford has a domestic reach rather than regional focus. It attracts many students from out of state/abroad and grads are mobile. Look at career report and many grads work in the Midwest, DC and East Coast.
Fels is worth considering and strikes a good balance btw theory and practice. Fels is also a smaller, more cohesive program with easier access to profs than a mega program with 400 students (SIPA). Philadelphia is an excellent laboratory for internships and capstone projects. Some students intern with the major's office or local non profits. There are plenty opps as many Fels grads work in local and state government. 
Just to round up the discussion, these 2 MPPs with education focus are located in the South but merit some credit.
UVA: Joint MPP/MEd
Vanderbilt: Peabody College's MPP in Education Program. Three concentrations right up yr alley: K-12, Higher Education Policy and Quantitative Methods in Higher Ed Policy. Vanderbilt has one of the best Education departments in the country (see ranking)

So there's a lot to like about this profile. I would tighten up the long-term goals but assuming decent execution, would be surprised if you don't get into some great schools.
I'm a huge believer in the power of networking so with your goals in mind, would submit apps at the following:
Harvard GSE: best education school, Harvard pedigree, networking opportunities with HBS/HLS/HKS, just 1 year program, many school senior leaders have this degree HKS: See above. A bit more flexibility than HGSE if that matters to you because not limited to education if you decide to pivot down the line Stanford: 2nd best education school, Stanford pedigree, networking opportunities with GSB/SLS, some cool nonprofit board opportunities in that area; Not as strong network in East Coast but Stanford degree will get your calls/emails returned Columbia SIPA: networking opportunities with CBS/CLS, lot of experiential learning opportunities with charter networks or the approximately billion nonprofits, opportunity to take classes at Teachers College UPenn Fels: similar reasons as above (networking w/Wharton, Penn GSE; experiential learning opportunities in Philly) but prestige factor falls off compared to schools above NYU: like Columbia SIPA but weaker reputation. Pretty sizable network in NYC though

The Sanford degree travels pretty much anywhere domestically, from what I understand just a step below the elites like HKS, Harris and WWS.  Stronger on the east coast for sure with a strong network in DC -- basically like the east coast version of Goldman.  I'll be fine wherever I want to go.

http://fordschool.umich.edu/careers-internships/graduate-students/jobs
This is a link to the post-graduate outcomes for Ford. I would think that the schools that you listed have the same (or similar) information on their sites. 

I was hesitant about the HGSE Masters for that reason - too many people go there. I think they accept 800 people every year (every one who applies?), for their one-year Masters. Doesn't this saturate the market? Still need to do more research here, I confess. I've only just started to look at MPP programs. That said, if I could combine a Harvard MPP with their Masters in Education, that would be something!

An acquaintance has same interests (domestic education policy and k12) with a quantitative focus. He did his MPP at Ford, he chose Ford as the MPP has the best fit for him (faculty research, internship opps, alumni, quantitative training). He got some good funding and there are opportunities for some TA/RA positions for qualified students. Two Ford profs are quite well known in education circles and publish in peer reviewed academic journals and NYTimes pieces on education.
I am not a Ford student.
I think Penn Fels offers some education electives and works with Penn's Grad School of Education.
 

That's a great list. I would add one program to it - Harvard's Master's of Education. I know a lot of people with your profile who have gone there. It's not as powerful as an MPA from Kennedy, but would send a strong signal of your expertise in and commitment to education that would be helpful for employment purposes given that this is the field you want to be in.

All the school's you've listed are the ones I would suggest for non-DC East Coast.  Sanford is known for its education faculty, and Harris has some good ones as well.  I'm not sure about HKS or Ford.  You might check into NYU -- it's an MPA but I'm not sure how education-focused it can be.
I only have one other suggestion, and it's overstepping my bounds a bit but I'll go ahead and say it anyway because I feel strongly: unless you want to help dismantle public education, please don't work in charter schools.  There are good ones out there, but the movement itself is absolutely a Trojan horse for school privatization.