I'd bet a considerable sum that she was the OW - or at least on the scene in some way. Probably she was the thing that made him make the break at least - as they say, men rarely jump unless they've got a soft landing lined up...
Your posts are really sad - you're trying hard to be loyal to your brother, but it's clear as day that he's acted, and is continuing to act, like a monster to his now-broken family. 'Can't get rid of her quick enough'? For heaven's sakes, why? He's got what he wanted, can he really not be as caring and civil as possible to the mother of his children, the person currently trying to put them back together while being smashed to pieces herself?
OP, just because he's your brother, does not mean that your own personal integrity has to go by the wayside. Yes, you can spell it out to him that while you respect his right to end the marriage and move on, and are prepared to support his new relationship completely without prejudice, he in return has to be realistic about the situation - and the fact that the people who should come first in this are the children.
People generally say that good parents should wait around six months before introducing children to a new partner after the stress of a breakup. Tell him you don't know how long he plans to wait, you hope it will be a decent amount of time and done in cooperation with your SIL, and he can see why, with all that uppermost, your instinct is to say that no, of course this new woman shouldn't be involved with the family before his own DC have met her. Has he thought for a minute the resentment and trouble that might cause?
The bottom line is, if he wants the new girlfriend accepted wholeheartedly, without resentment, he has to let it happen at a pace other people are happy with. Properly happy, where they can see they aren't causing pain to other loved ones. It just takes time, and he has to suck that up. Force it, try and push it, and the new gf will be the one resented. Probably most of all by his children.
So refuse, tell him why, and make it clear that you are actually genuinely trying to make life smoother in the long run by doing so.