I must say, the question is backwards.
It is the responsibility of the person who claims that “the priest must mention the name” to cite any law (canon law, liturgical law, etc.) which supports that claim. Simply put, there is none.
Along with what Don Rugerro has already posted, I’ll add this:
Mentioning the name of the person for whom the Mass is offered is done in the Roman Canon, which has places for the living and the deceased.
In EP II and III, mentioning the deceased by name is strictly an optional addition. There is no place to mention the name of the living.
In EP IV, there is no place to mention either the living or the deceased.
Now, according to what your friend claims, if someone requests a Mass intention for a deceased relative, and the priest schedules that intention for a Monday in Ordinary Time, then chooses to use EP IV for that Mass, the intention would be “void” and it would not be fulfilled. That makes no sense.
Again, it’s your friend’s responsibility to support such a claim. I can assure you that it isn’t possible.