MBA vs MS Finance

Hey guys,

This topic title is fairly common and there are many interesting viewpoints to this dilemma many people face.
A little bit of a background on what I am.

Final year undergraduate in Biomedical Engineering from a reputed University in India, but a non-target university to most programs abroad.
8.6/10 cumulative grade point.
Tonnes of leadership experience right from high school (Student Council Leader thrice, club/chapter president, sports team captain)
Extracurriculars, mostly in music, debate, theatre and sports.
A little bit of community service.
Internships in the undergraduate field of study as well as one at an investment firm.

I will also be doing another internship in a few months. I have the option of doing another finance internship in the industry, but I would prefer an academic internship as I feel it provides more depth to my profile and more credibility to whatever knowledge I will claim to possess.

I aspire to be an investment manager.
I will be giving the Level 1 CFA exam next June, immediately after I graduate.

The questions I have are -

1) Considering the fact that my engineering stream (and my university) makes it very hard for me to land a decent job in an investment/finance capacity, would it make sense to try and pursue an MS in Finance.
The reasons I have for this decision are -

a)Branding, from a good university so that I can achieve an MBA from one the top B-Schools in the US (This branding I lack at the moment, as unfortunately, like millions of other Indians, I am not from the prestigious brand of an Indian Institute of Technology)

b) Inability to land a job given my biomedical background. It is hard to land a core job due to the biomedical field being new. And to be honest, the prospect of having to do a few more years of biomedical really displeases me.

c) The MS Finance seems to be a good finance degree to have due to the options it provides, that the CFA does not. (I may be wrong about this). I also know that an MBA after Mfin might be redundant curriculum-wise.

2) Should I try to land a mediocre job which may or may not align with my interests, pursue my CFA on the side, and try to glean as much experience as a good B-School might require and directly apply for an MBA?

My reservations with this approach stem from the belief that, 'the better the last thing on your resume, the higher your chances of landing something like an HBS'
Once again, I do not know how true this is, but I wouldn't be entirely wrong in believing that the more reputed my employer, the better my chances.
And given the difficulty landing a job immediately after my engineering, I am not entirely sure this approach is worth it.

Sorry for being really wordy. It sounds more like a rant at points, I know.
I would really appreciate some help.
Thank you