Is the husband entitled to obligate his wife to bear the responsibility of his daughter from a previous marriage, and would the wife be sinful if she refused, knowing that this is not religiously mandated for her?

A wife is not obligated to serve her husband's children from another wife, nor his relatives. The husband does not have the right to compel her to do so. If she does, it is a kindness from her for which she is to be commended. Obedience to the husband is not obligatory in all matters; rather, it is restricted to what pertains to marriage and its related affairs, and to what does not involve disobedience [to Allah] or harm.