Get experience or wait out current company?

Currently working in back office role at larger PE firm while pursuing MBA (top 10 part time). Want to transition into acquisitions/asset management. Been approached by some smaller shops, introduction phase so not sure on pay or anything. Curious on people's opinions. Stay and try and leverage the fact that my current company knows me (risk they won't hire me on otherside of business) or leave to get experience in what I want to do while I finish my degree? Seems like people like MBAs, but prefer experience.