How can one deal with a wife who became pregnant through adultery, claims repentance and remorse, and the husband refuses to separate, out of consideration for her psychological state and fear of exposing her, while he doubts the lineage of the fetus despite his knowledge of the hadith: "The child is for the bed"?

If you are reasonably certain of your wife’s sincere repentance and no doubt has emerged from her, then keep her and do not divorce her. Sincere repentance erases sins. Be keen to start a new chapter with her by strengthening the relationship, keeping her away from sources of temptation and doubt, and shielding her. The child is legally yours, and it is not permissible to deny his lineage through DNA testing or any other means. Consider what has befallen you a trial, and hold yourself accountable and renew your repentance, for trials do not descend except due to a sin and do not depart except through repentance.