All perfect praise be to Allah, The Lord of the worlds. I testify that there is none worthy of worship except Allah and that Muhammad is His slave and Messenger.
If the wife is not persecuted in her own country, then it is not permissible for her to lie to the government in order to obtain asylum, and it is not permissible for anyone who is not persecuted to lie to obtain asylum. If anyone lies, it is not permissible for him to take the benefits assigned for refugees because this is taking money by way of lying.
If you mean that the wife did not lie by saying that she is persecuted but she is banned from traveling as long as she is given asylum because her husband got asylum and if she travels and informs the government about her travel and this leads to the withdrawal of the right of asylum from her and she will be expelled from the country and will be separated from her husband and children, then there is no harm in her not telling the government about her traveling in order to ward off harm. Indeed, the scholars stated that lying for a preponderant benefit is permissible.
As-Saffaareeni, from the Hanbali School of jurisprudence, said in Ghithaa’ Al-Albaab, “The adopted view of the (Hanbali) School is that lying for a preponderant benefit is permissible...”
Imaam An-Nawawi said in Riyaadh As-Saaliheen, “Know that lying, though forbidden in principle, is permissible in some cases, with the conditions which I clarified in the book Athkaar; the summary of them is that speaking is a means to achieve objectives; any praised objective that is possible to obtain without lying is forbidden to obtain by lying. But if obtaining it cannot be done except by lying, then it is permissible to lie.”
There is no doubt that not ruining the family is a preponderant benefit and includes a praised objective; so she is not obliged to inform the government about the truth of the matter in order to avoid harm.
However, she is not permitted to take the benefits given to her because it is money taken by lying, unless there is a dire necessity, such as if the spending of her husband on her is not sufficient (to cover her basic needs) or the benefits that he gets are not sufficient and he cannot get a job, and so on.
Allah knows best.