Solost - Your H is being a cowardly knob - he is not worried about damaging the DC, he is worried about damaging HIMSELF in their eyes! What a prick! Does he really think he's going to get away with this?
Your H does not get to call the shots on this! He is not there to pick up the pieces, YOU are!
However, you are in a pickle because you still haven't seen a solicitor yet, and your H coudl turn nasty if you go against his pathetic wishes and talk about BB.
It looks to me as though he is going to make out that you and he have split through differences and that later he will claim to have "met someone else" so it looks like there was a decent break - the fucker!
I'm hoping that your oldest child at least will ask awkward questions that your H is spectacularly unable to answer - and he will end up lying. at which point, I think you have to say "you must tell them the truth!"
I really feel for you, this is so shite - I hope someone with actual experience of this can tell you it will be ok, but I am very worried about his deviousness. And what will happen if you undermine it.
You have to keep trying to explain to him that his LYING to his DC is going to be far more damaging to them than them being told the true state of affairs. you can also throw in a comment about "it's not like it's going to change any time soon, is it, so what's the point in delaying?"
(((Hugs))) to you - I am so :( for you in this situation.