I didn't realize it the first time I saw the movie. But it's clear in that convo on the phone when he's dragging Toby with him that he's not telling the BBC not to run the story, he's actually "accidentally" leaking it to them in the guise of asking that it not be run. He wants to use the story as an excuse to sack Simon.
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I agree, I thought they made this quite clear in the film - you see a 'light bulb' moment during his close-up in the meditation room, and his eyes show he's formulating a plan. He then immediately calls the BBC, sounds insincere when pleading withthem not to run the story, and then as soon as hangs up, he triumphantly says something like "and he's gone" or "and he's taken it."
Not only does he want to use the story as an excuse to sack Simon, but he mainly wants the BBC to focus on the wall story and not the war story. He's deliberately leaking 'slightly bad' news to throw them off the trail of the 'very bad' news.
he mainly wants the BBC to focus on the wall story and not the war story.