If you already hold the CFA, FRM, and CAIA designations, you are going to get very little return on your investment if you pursue an MBA. First of all, achieving all three of those by age 24 means you have been studying and testing like crazy for the last three years (achieving all of these by 24 is actually somewhat skeptical IMO). Do you want to spend almost all of your 20's in one form of school or another?
An MBA may be, more or less, required for IB, but it certainly isn't for PE. Analysts and managers in PE span the entire gamut when it comes to education; BS/BA only, BS/BA + CFA, BA/BS + CPA, MBA, BS + JD, BS +JD/MBA, etc... it's all over the place (but very very few MFEs).
Take a bit of time to explore what options you have right now. Spending another 50-150k on an MBA probably isn't worth it with respect to knowledge, skills, opportunity costs, and future opportunities/salary.
I have actually begun to notice some firms (buy side) looking down upon candidates who have degree after degree and designation after designation. If they are looking for someone to help make solid investment decisions, personal actions/investment (like being 100k+ in student loan debt) are somewhat indicative of their skillset/judgements. Personally, I have come to the conclusion that after one or two degrees/designations, you get almost 0% return on the investment. That said, you never see diminishing returns on experience and demonstrable skill (which may or may not come from any degree/designation).