"The Catholic position" does not hold that when the Bible refers to the brethren of Christ that they are his cousins. The Church holds only that Mary did not have any other children besides Christ. Who the brethren were is debatable--they might have been cousins (this is the most common view today), they might have been stepbrothers via Joseph (this was the common view before St. Jerome), or they might have been adoptive children.
The premise of the argument--that the New Testament says Elizabeth is Mary's cousin--is wrong. The translation being quoted does not accurately reflect the Greek.
The New Testament does not say that Elizabeth is Mary's cousin, the Greek word for which is anepsios. The word used in Luke 1:36 to describe Elizabeth is suggenes (pronounced su-gen-ace), which simply means kinswoman or relative. It tells us nothing about her exact relation within the extended family. All we can tell from the word suggenes is that Elizabeth was some kind of female relative of Mary's. But whether she was an aunt, a cousin, or a more distant relation cannot be determined from the word.
In a few places the New Testament does use anepsios, but this does little to argue that the brethren of the Lord were sons of Mary. Arguments from word choice (i.e., "Why this word instead of this other word?") are rarely decisive. New Testament word choice is especially difficult to build arguments from since it involves a mixture of Jewish and non-Jewish word preferences. The brethren of the Lord may have been brethren of a different kind (e.g., adoptive brothers or stepbrothers) without being half-brothers through Mary.