"Live as brother and sister, help raise whatever amount of children they have both had together".
I am not clear on how this works in practical terms. The woman is not your sister. She is someone you slept with.
Why is it less of an occasion of sin to share a home with your new "wife" than it would be for a widower to move in with an ex-girlfriend to help raise her kids?
I think you confuse what I am saying, I am not suggesting they both necessarily have to live together in the same home. Maybe they could build a separate room if possible, talk to the children about it.
Yes, there is of course the proximate occasion of sin. The main point is that you already do have children, and those are an obligation that they have to keep now. There is no doubt about that, at this point it is best not to re-marry. If someone is truly sincerely repentant for the mistake, etc... I am sure they would make the right choice.
Really depends how big your home is, its not that impractical. Most modern couples when they are older aren't that intimate so if this is late enough, it would be much easier then if you are talking about two young people in their 20's. The more children you have had, the harder it is. If you have had no children, then it can really be much easier to break it off without having to worry about the repercussions to any children.
My parents aren't married and haven't been together since pretty much my birth. Yet for the past 12 years, they both live together in the same home. Of course this situation is totally different. However, there are many similarities they both have several children they both engendered. So it can work, if just one of the couples is Catholic. Doesn't work perfectly, and it can certainly be psychologically distressing knowing your parents are not sharing the marital bed, but at some point you get over it and see the objective truth. Whether or not it is worse, for them to be "married" or separated. Ultimately in my specific case, I am glad that they never got married, and in a certain way this was my actual defense from the Novus Ordo. God certainly works in mysterious ways, for I would rather have an irreligious mother, who is not a Catholic, then one who is a religious heretic. She could then exert moral influence over me, while in the other case, ever since I received the gift of faith (I was taught the faith/catechism) I saw that I could not obey my mother in those things that contravened Divine law/ecclesiastical law etc... In many ways this was part of the reason, that made me such a strong Catholic, I had to constantly defend my faith and research/find what Holy Mother Church had to say.
I could certainly empathize with those children (since I know from first hand experience what it is like), but what a great gift it would be for them to realize the objective good that it would bring for their parents to separate, because what God has put together let no man put asunder. If for example, the impediment to the marriage stops. I.e. that the other couple dies where the annulment was given, then they could be able to then licitly marry. It would of course be evil, to pray for their sudden death so you can remarry, but we have to follow Church teaching, no matter how difficult it might be. Many times, some of these people truly have overpowering feelings for one another. One particular case reminds me of Padre Pio, when he was confessing a notorious sinner (an actress I believe) who was living in adultery with a married man. He told the woman, that he was going to die soon and that if she truly loved him she has to separate from him. He told her, that he would save his soul if they separated. She followed his advice, got absolved and obeyed. Yet, both of them really wanted to be with each other, but I am sorry. What God has put together LET NO MAN put asunder, no exceptions. This is what it means to carry your cross, sometimes through our own fault we make such serious mistakes that they affect your whole life. This is what it means to be penitent, whereby you are willing to take responsibility and to the best of your ability restore as much of the initial damage done as possible.