In addition to the requirement that she “declare that [she] is prepared to remove dangers of defecting from the faith,” the Code of Canon Law requires your cousin’s bride-to-be to “make a sincere promise to do all in [her] power so that all offspring are baptized and brought up in the Catholic Church” (CIC 1125 §1). Your cousin, on the other hand, need only “be informed at an appropriate time about the promises which [she] is to make, in such a way that it is certain that [he] is truly aware of [her] promise and obligation” (CIC 1125 §2). He is not required to make the same promises. In fact, the marriage can go forward even if he is resistant to her promises.
The New Commentary on the Code of Canon Law explains,
Permission for a mixed marriage can be granted even when it is foreseen that the Catholic’s efforts to pass on the Catholic faith will probably be fruitless because of the resistance of the non-Catholic spouse. In these circumstances, the Catholic party can fulfill his or her obligation, at least in part, by playing an active part in contributing to the Christian atmosphere of the home; doing all that is possible by word and example to enable the other members of the family to appreciate the specific values of the Catholic tradition; taking whatever steps are necessary to be informed about his own faith so as to be able to explain and discuss it with them; praying with the family for the grace of Christian unity as the Lord wills it. Doing all that one can does not include so insisting on the Catholic formation of children that the stability of the marriage is threatened. What is necessary is a sincere promise by the Catholic to do all in his or her power to assure the Catholic formation of children. The sincerity of that promise is to be presumed, unless there is evidence to the contrary. (1346-1347)