嫁 in context: a daughter-in-law and a wife back home
Compare a mother cooking with her daughter-in-law with a man writing to the wife he left in his hometown. The relationships in each scene determine which meaning of 嫁 applies.
A daughter-in-law cooking with a mother
嫁 (よめ) can mean daughter-in-law. In “母は嫁と一緒に夕食を作った。” (“My mother made dinner with her daughter-in-law.”), the speaker's mother made dinner with her daughter-in-law. 母 and 嫁 are the people participating in the cooking.
Calling 嫁 the mother's wife would change the relationship in this example. The scene identifies a mother and her daughter-in-law cooking together; it does not describe every family member taking part.
A letter to the wife back home
In “物語の男は故郷に残した嫁へ手紙を書いた。” (“The man in the story wrote to his wife, whom he had left in his hometown.”), 嫁 means the wife of the man in the story. He wrote to the wife he had left in his hometown, and 嫁へ identifies whom the letter was addressed to.
故郷に残した supplies the wife's background in the hometown. Writing the letter does not establish that she returned or that she answered it, so keep the reported action with the man who wrote.
Follow the relationship in each sentence
The examples share the spelling 嫁 but describe separate scenes: a mother cooking with her daughter-in-law and a man writing to his wife. Keep the relationship to the mother or the man with each meaning.
一緒に identifies the person cooking together with the mother, while へ identifies the recipient of the letter. Those surrounding expressions help prevent a single fixed interpretation of 嫁 across both scenes.
Is 嫁 in 物語の男は故郷に残した嫁へ手紙を書いた the man's daughter-in-law?
Show the answer and why
No. Here 嫁 refers to the man's wife, whom he had left in his hometown. Keep that letter scene separate from the mother cooking with her daughter-in-law.