MBA or MFin after 3 years of banking?

Hey guys, long time lurker on the forum here - been following since freshman year in college. I now have 3 years of experience working as an IB analyst in a coverage group at a tier 1 bulge bracket bank in NYC. Due to my visa issue, I unfortunately have to pursue a master’s degree to come back to the U.S. next year.

I am in the process of deciding whether I should apply to an MBA or a Master of Finance Program. I am currently leaning towards a one-year Master of Finance Program because it’s shorter, more cost-effective, and can allow for 3 chances with the work visa lottery (H1b), while an MBA only gives 1 chance with the lottery.

Question 1: My goal after graduation is to get an Associate position doing corporate development, corporate strategy or FP&A etc. I understand that generally an MBA degree is perceived more favorably, but given that I have had 3 years of experience in IB, do you think my chances with landing a corporate dev gig will be jeopardized by the Master of Finance program (rather than the MBA)?

Question 2: I have been in regular contact with headhunters during my 3 years of banking career. If I am still interested in Associate positions at smaller PE funds / investment firms after graduating from a Master of Finance program, will I still stand a chance with these firms through headhunters? I understand that they typically work with current analysts but, again, given that I have the right experience, will I still be able to at least generate some leads through their help?

Thanks!
smalltown