Brand Management from Non-Elite MBA?

For anyone here who has entered Brand Management, how do you say is placement not coming from an elite (or even Top 50) MBA program (for top companies such as J&J, Unilever, Mars, or P&G)? Marketing professional with 5 years experience (1.5 years in Management so far), and graduated from a large public Uni. My experience could enter me into an elite school, but I would rather avoid taking loans (or even lose out on pristine experience) and do my MBA part time. Call me old-fashioned, but I can't stand the thought of taking on debt for an education. Plan on taking online courses at Illinois for cheap and transfer them when starting in the Spring.

Why do an MBA and go for Brand Management? To be frank, I could use experience in project and team building. I manage a very small team, and could always use more experience. My experience is good and my career is moving well. (Mostly) like what I do and I do it really well. But I am specialized into a niche part of Digital Marketing (which can feel limiting at times), and I just want to do branding. Working with consumer products is my dream. Would have pursued that if I knew this role existed in college.

Since I live in NYS, I'm deciding between either Rutgers, Baruch, and Buffalo (far but I like the area more). My only concern is that I think all 3 schools focus a lot on Accounting (and Supply Chain for Rutgers/UB). In general, only Midwest or expensive schools are strong in branding. In Buffalo, I would be at least close enough to the Midwest that I could apply for positions and internships in Indianapolis/Chicago/Michigan where most of these positions lie (though I would need to take off work to pursue that internship).

Does pursuing an MBA at a large business Uni hurt my chances of getting into BM? Of the 3 unis, which one (if any) presents the best opportunity, both in the short-term and long-term?

Would I need an internship as well to get into these companies? If so, is 1 sufficient or should I aim for 2? As an intern, can you apply for places that are far and simply relocate temporarily? What's the protocol for a part-time student doing an internship (ie: work then quit their job)?

Thanks for anyone answering this question! You'll forever have my kudos