It turns out that when Rod Rosenstein announced the indictments yesterday of 12 Russian military intelligence officials, he said that responsibility for prosecution of them was transferred from the Special Counsel's office to the national security division of the Justice Department. While it’s unlikely that such a trial would ever go forward, given the profound unlikelihood of their being extradited, what this means though is that since this division is now officially handling it, even if Trump was able to shut down the Special Counsel's investigation, this would still go forward.
But more than that, it would be difficult for the White House to gain much access to the material, given what the department is. Someone described this division as being “a top secret part of the Department of Justice. Mortals are not welcome: only spooks. They'll compartmentalize it, and even Trump would be blind. Even Trump's new head of the DOJ's Criminal Division -- Brian Benczkowski, with ties to a Russian bank -- who Republicans confirmed on Wednesday, wouldn’t have access to it.
As has often been said, while the Trump administration is playing checkers (or at times, it seems, tiddlywinks), Robert Mueller is playing chess.