How would it be disastrous for his job? I realise you might not want to be specific but can you give us a clue?
Does it say anything libellous about him or is there a court order banning identification of the children?
Does it say anything about him at all other than he's her ex-husband and the children's dad?
If it's just that he doesn't want it in there I don't think there's anything he can do and I don't think there should.
Lots of men got injunctions to stop ex-girlfriends telling their stories simply because they had the money. Many of them are still in place. In many cases it is an abuse of power. It's too easy to say these stories shouldn't be printed because people just want to gossip. Sometimes they shouldn't, but sometimes they definitely should.
Fred Goodwin was able to get an injunction banning talk of his affair with a senior RBS employee which may have been relevant to the fuck-up at the bank. It's still in place for her.
There's also shocker where someone involved in football got an injunction gagging the husband of the woman he was shagging on the spurious grounds that it might drive his own wife to suicide.
And the actor who got his co-star dropped from the show after he'd finished shagging her on the grounds that his teenage children might be teased at school.
I'd have thought her right to expose dodgy employment practices at a major TV company would have over-ridden that.