This is what I don't understand.
What's so difficult about saying: "Look Mate, you're busted, and we know where your kids go to school. But as long as you tell Ronald McDonald that the protest is next Thursday instead of Friday, we'll pretend nothing ever happened."
Well of course I know what's difficult about it--you have to be a complete bastard to say it. But it's not practically difficult is it?
Well, there's a few minor problems.
First of all, your agent goes back to his principals and says "these people are threatening my family", and his principals, if they have any interest in their spy continuing to operate as such, take the necessary steps and instruct him to say "Dunno what you're on about, guv"; either that or they decide that their man is blown and pull him out.
And you need resources. Lots of resources. Years ago, I used to work for a firm that did a little work for a firm that did a little work for a firm, etc., that did interesting things with surveillance and so on: if I learned anything (given that I wasn't anywhere NEAR the interesting stuff), it's that with anything covert, you basically start off by multiply the manpower by 9 - case handlers, liaison, communications, all that stuff. All this stuff of James Bond wandering off on his own is fairy story stuff. And, while I don't imagine police spies are particularly concerning themselves about co-ordinated anti-espionage stuff, it's going to be built into the system with all kinds of safety mechanisms.
You'd essentially need to match that, and then some, to set up and run a double. The motivation would have to be there, too - no point blackmailing them, this would be about ideology, not threats, it's a copper, remember - and they'd need to be VERY sure that you could look after them, especially given that they've already demonstrated that they could penetrate your operation. Just supposing you got a live one, he'd been converted to your cause, and was tempted to double, HE is going to need to be sure that there isn't someone else in your operation who's reporting back and who could burn him.
It just wouldn't be worth the bother.