I don't think the Church could outright support it, since it would not be a matter of faith and morals. What must instead happen is condemning fiat currency as a theft by a government of its people.
One must admit that the Catholic Church is the most influential single organization in the world since 1 of 6 people on the planet is technically Catholic (technically, but less practically). If the Catholic Church can condemn fiat, the ripple effect would be enormous. Other faith communities would soon follow.