Virtually every article that I have read on this topic says they can request the courts to put this on hold. Mueller isn't above the law, areas of gray are interpreted by the courts. Mueller is taking the approach of asking for forgiveness before permission. It was a non-transparent approach to getting the emails. Why did Mueller handle it that way? What was he afraid of? Getting rejected?
This is exactly what the courts are meant to resolve.
So why haven't the transition team lawyers done that? Because it's settled law. The transition team had no special privileges, certainly not in writing. It would probably be illegal to grant any.
They can't claim executive privilege because Trump was not yet the Chief executive & they can't claim attorney client privilege because of lack of due diligence on their part. After the buttery males incident, everybody & their dog knows or should know that there's no privacy on .gov servers.
Why did Mueller do it that way? Because the law grants him that ability, plain & simple. When investigators have information suspects don't realize they have then investigators can lead the suspects into demonstrable lies. It's a common technique.