Is it permissible to apply for American citizenship as a precautionary measure to protect a child in the event of marital problems, knowing one's religious responsibility towards the child?

It is not permissible to obtain citizenship of a disbelieving country unless there is a necessity or a need deemed valid by Sharia (definition). Similarly, residing in non-Muslim lands is not permissible except under certain conditions and regulations. It is not permissible to apply for citizenship and commit Sharia violations to gain a perceived benefit or avert a perceived harm. It is advised to return to a Muslim country to live with dignity, far from the corruptions of non-Muslim lands.