Is the presence of the divorced woman in her former husband's house after Khulʿ, to care for her children who chose her, considered a sin, in light of the absence of alternatives for accommodation and child care?

The woman who has obtained a khul‘ (divorce at the wife's instance) from her husband becomes separated from him and a stranger to him (ajnabiyyah). Therefore, it is permissible for her to live with him in the same dwelling, provided that her room is separate with its own facilities, or that a mahram is present with her if there is no locked door. However, if a separate dwelling is not available, it is not permissible for her to cohabit with him except in cases of extreme necessity, while taking precautions to prevent fitnah (temptation/strife). This necessity ceases when its causes are removed.