Which should take precedence, Hajj or Umrah, given the wife's insistence on Umrah and the inability to perform both?

Hajj is an agreed-upon pillar [of Islam], while there is a difference of opinion regarding the obligation of Umrah. If Hajj and Umrah conflict, then prioritizing Hajj is the imperative, and it is possible to combine them. It is not permissible to miss Hajj for the sake of Umrah, unless the Umrah is performed before the Hajj season. If the Hajj season arrives and one is unable to perform it, then there is no blame upon you, as Allah Almighty says: "And [due] to Allah from the people is a pilgrimage to the House - for whoever is able to find thereto a way."