If he wanted to leave her, he could do it in a heartbeat. His parents would be overjoyed. He will assume the kids and you would be overjoyed. His friends would all congratulate him on coming to his senses. Work would be easier, as it doesn't sound like they approve of this sort of thing. He'd have so, so much more money, if he had only one household to run.
Please, please, stop listening to what he is saying, because it will always be, to an extent, what he thinks you want to hear and what serves his interests. Instead, look at what he is doing.
He doesn't love any of you - not BB, not you, not the kids - in any adult, unselfish way. He's acting only in his own interests from start to finish - he's not put ANYONE first except himself at any stage of this.
He is with BB. And given the advantages he'd get from coming back, he is with her because he wants to be. Not because he feels he owes her (didn't stop him from leaving all of you when he owes so much more, nor would it stop him going back if he wanted to) and not because he has no choice. He very, very apparently believes that he does have that choice.
You deserve someone amazing. You so, so do not deserve to be this man's secondary consort, like he's some 19th century Chinese Emperor. The only reason you aren't still sleeping together is because you called a halt - he has two wives, effectively, and he's living with the one who makes him feel like he's a young professional, while the family wife picks up the domestic slack.
BB is indeed a bunny boiler. But she's also a childless woman pushing forty, who seemingly longs for a husband and a family, who is being made an awful lot of promises by a guy who doesn't seemingly plan to keep them. If she isn't trying to get pregnant, she may discover too late that a family with her is not his plan, and she won't be of an age where a family with anyone is an option. Now, no reason at all you should give a shit about that. But he is supposed to love her so much it was worth leaving his own children for - and this is how he treats her? This supposed love of his life? And he behaves to you in a way that is that disloyal to her, just as he has betrayed you with her? I think that speaks volumes.
I am truly so very sorry to say this. I am not saying it to hurt you, and I hate to think that it may. But he is playing you, and you could still be locked into this game in ten years' time if you aren't careful. My aunt was. She was in her 50s before she was emotionally free of her ex. These are months and years you will never get back - and you could be spending them on someone worth your time and thought and care: yourself, your kids, and one day a nice man. Because they are out there, and they would treat you as the lovely, strong, kind, interesting woman you are - not as someone to keep warm on a back burner.
You are such an amazing, courageous, graceful and dignified person. It's so apparent from all your posts. He is insane to have made his choices - but he has, and just as importantly, he still is. And I so, so wish you would make choices by seeing what he is doing, and realising that is who he is. And what he is is someone who is not, and probably never was, good enough for you.