This whole article starts out with a historically-correct account of the people associated with Triumph, and as livius states, tries to distort the truth and discredit modern-day conservative Catholic leaders.
As for what livius says about the TFP ("TFP... has no official Church support and little or no unofficial support, either."), he isn't too far from the truth. They definitely don't have the resources of Opus Dei.
Earlier this year, they had a large campaign against the Da Vinci Code movie, and I participated in their street protests in the DC area. A bunch of bishops supported their campaign as well. Since then, they have had a few "e-protests" against anti-Catholic activities across the country, and I signed the associated petitions. They're among the very few (including the Catholic League) that draw attention to such activities. I've heard the wacky things about them as well, but they don't seem to have much documentation. If it's true, why would Archbishop Burke and Fr. Trigilio have anything to do with them?
I had the honor of meeting Fritz Wilhelmsen at a seminar at Steubenville in 1990 on, yes, the thought of John Courtney Murray. During a break I asked him if the red berets they wore that day had anything to do with Carlism (I already knew that he taught a course each year at the Escorial). He drew himself up to an imposing height (he was taller than me, anyway) and said: "Why of course. I am an honorary Requeté you know." He was a legendary character.
His daughter, Dr. Alexandra Wilhelmsen, has actually pubished some very interesting articles on Carlism, pointing out some of its more attractive features as mentioned in part in this thread. To me it was especially interesting to find that Carlism was a monarchist movement that recognized that a monarch could in effect abandon his right to the throne by failing in certain respects in doing his duty, such as to uphold the fueros, as I recall it. "Pure" monarchists would say that the king could never abandon his rights, even through abdication--I think this is the Habsburg family's position, in effect denying the validity of the abdication by Blessed Charles of Austria.
One or two Carlist pretenders when making an incursion into Spain swore the royal oath to uphold the fueros (I believe this was done even at the famed tree of Guernica, where the old rulers had sworn the oath and which was what made Guernica significant in the first place, but which the Nazis were too clueless to know to destroy in their air raid).
This whole subject is part of a much larger historical topic that seems largely unknown today, the extent of local and regional liberties in pre-Revolutionary Europe. To juxtapose the thoughts here, I first had my consciousness raised on this point by an Opus Dei priest and himself famous University of Navarre historian, Fr. Federico Suarez, writing in a festschrift book for Fritz ("Saints, Sovereigns and Scholars" [I may have reversed the order, though]), in which he noted in passing that monarchical central governments interfered less in the daily lives of the people than post-Rveolutionary governments do. His cite was to a very interesting book by Frantz Funck-Brentano, entitled something like "The Ancien Regime in France." That book is itself very worthwhile reading, and really eye-opening in a way. If so moved, some day I will cite passages from it.
Fritz wrote articles for the SSPX's "Angelus" magazine in his later years, and Alexandra told me that when she told him that he would be thought to have gone over to the Lefebvrists (he in fact was an adherent of the FSSP's Indult Mass in Dallas), he told her "most people ecumenize to the left, but I ecumenize to the right."