From what I've seen, firms generally won't go to hire some one out of an MBA program to run a 1000 person division. Seriously, a 1000 person division being run by the typical newly minted MBA with 3-5 years of experience (I realize you mentioned 7-8, but that's more an outlier than the norm)? The typical FT MBA student has managed some where in the 0-20 people range...that's a big step from 500-1000. That said, many firms will go to MBA programs and hire individuals who they expect to be running 500-1000 people teams in 5-10 years. At these firms, frequently the only way to get into such a leadership role is to have gone through the MBA program and then get hired into their GM programs.
That said, you have to be realistic in your expectations. If you're already running a group of 100+ people, chances are a FT MBA isn't the best choice for you; and EMBA or Sloan Fellows program is likely the better choice.