Is the husband obligated to accept his wife's brother's offer to pay the rent for an external apartment for one year, given that the wife is bargaining for divorce or external housing, and she refuses to live in a separate apartment on the second floor of the family home?

The wife has the right to an independent dwelling that meets the Sharia specifications. If such a dwelling is available, she is not entitled to request another, and the children's welfare does not justify it. There is no harm in accepting her brother's offer if she does not fear his obligation and hopes for benefit. Strive to reconcile between the wife and the family if there is a misunderstanding.