At the moment, Rosenstein is supporting Trump and Sessions, quite well. Who knows where his true allegiencdes are. At his level of promotion (#2 in the FBI), all of these career government types are self-serving snakes.
As for "why Mueller?" maybe Rosenstein has judged him to be reliably fair in this case (not necessarily in general), or he may know that Meuller has a dirty history, and a possible way to get revenge on Mueller is to give him this assignment, hoping that Mueller disgraces himself with obvious partisan output.
Thanks !