Would the husband be sinful if he separated from his wife, who betrayed him via mobile messages, due to his discomfort with her and his inability to provide two separate residences for them?

Divorce is permissible when there is a need for it, but disliked otherwise, and it can even be forbidden. If the wife corresponds with foreign men, she has committed a great wrong. If she repents and becomes upright, then keep her and do not hold her accountable. But if she persists, then divorcing her is recommended. However, if you fear a greater harm, such as the children being lost, then there is no sin in keeping her, and work on educating her. If she does not mend her ways, you may discipline her. As for cohabiting in the same dwelling for two co-wives with their consent, there is no sin in it. But if they dispute, then divide the house between them.