Congratulations on your acceptance! That is exciting. In terms of applying for your MBA after obtaining an MMS or MiM, you would need to clearly articulate why you need the additional Masters degree to pursue your desired career path. You would want to communicate what skills and experiences you already have (from MM and work experience) and what you still need to learn/experience from an MBA program to achieve your goals. Do you feel that there would be enough difference in the programs for you to do that?
Beyond that decision, it sounds like you need to better understand the progression of your desired career path. I would recommend that you speak with others who are in similar careers and ask about their career paths, education, etc. They may be able to provide insights if you need work experience or a Masters at this point.