You’d like to know whether there are any other universities that might accommodate cases such as yours? How about none other than arguably the most prestigious engineering school in the world: MIT?
Here’s a woman whose undergraduate degree was in English literature - a field surely farther removed than a social science undergrad degree relative to engineering - who nevertheless later obtained an MS in engineering from MIT. To be sure, that MS is in ‘Engineering and Management’ from MIT’s Systems Design and Management (SDM) program. Nevertheless, it’s still a bonafide MS degree awarded (jointly) by the MIT School of Engineering. Nor am I aware of MIT requiring her to complete any makeup work whatsoever. Nor does it seems as if she paying much of a penalty (if any) for not holding an undergraduate engineering degree, for she’s now the Director of Engineering at a software firm.
https://www.crunchbase.com/person/ilana-davidi-reeves#/entity
Want another example? How about Stanford? The Management Science & Engineering MS program - a degree awarded by the Stanford School of Engineering - specifically states that applicants need only have “completed a course in differential calculus of several variables before applying.” There is no requirement that applicants hold an undergrad STEM degree. Even if you’ve never taken such a course before, you can easily complete one at the University of Phoenix or other similar online entity.
https://msande.stanford.edu/admissions/frequently-asked-questions-graduate-admissions
Granted, none of these examples are regarding an MS in mechanical engineering specifically. The OP therefore needs to ask himself how important it really is for him to obtain a mechanical engineering MS. {Personally, I’d be far more inclined to obtain an engineering master’s from MIT or Stanford rather than from BU or NYU even if it wasn’t in my preferred field of engineering, but perhaps that’s just my personal preference.}