I had the same conundrum (MPH vs MSN-PH) when I was thinking about graduate school. Ultimately I went with the MPH because I believe it is more versatile - also, among the nurses working in public health that I know, it seems to be that the MSN-PH is more held by nurses who have been in the field for >15 years and that the MPH is the current trend for nurses doing their education now or more recently. This could vary depending on your region I guess, but that's what is true for my area. I can't speak to whether an MPH or MSN-PH would be more appropriate to a lactation consultant position, but if you know any nurses who do that for a living you could ask them what type of advanced degree (if any) they have. I would imagine that IBCLC certification is probably the most important credential you could have for that job.
I would be very leery of attending an MPH program that was not accredited, as it may limit you in future opportunities depending on the type of work you want to do in the future. For example, if you want to work for a federal office (CDC, NIH, Indian Health Service, etc) or use your graduate degree as a level up for a uniformed service officership, lack of accreditation could prevent you from having that credential recognized. This may not be as big of a deal to you if you never want a job like that, but consider that just as you would not want to attend a nursing school
that did not have CCNE or ACEN accrediation, you should probably apply the same standard to your education in an MPH program if that is the route you select.
Good luck, and let us know what you pick!