You said: What interpretation actually gives the Catholic Priests the ability to perform these miracles at will?
The premise of your question is wrong. I sense you are misinformed. Catholic priests do no perform miracles.
During the Mass (and Divine Liturgy in the eastern Church), it is Christ who changes bread and wine into His body, blood, soul and divinity, as He did at the Last Supper. It is a sacramental representation of the once and for all sacrifice.
I understand that the priests don't actually transform the bread and wine, but there is also no reason that Christ does so every time a priest does the ceremony. I don't believe priests as such are even necessary - heads of the local churches are to be teachers of the Word, that we learn it and have the opportunity to become followers of Christ and then to hopefully carry the Word to others. As I see it, the main purpose of churches, as congregations, is to serve as a place for us to gather in worship and learning and to help keep us focused on the Word. "Rabbi" means "teacher" and I feel that folks did too much reading into the Bible to come up with all the requirements laid out in the various religions. The Bible stands on its own.
God Bless