My problem with the K of C is that it is all about V2. They LOVE V2 and they loved JPII.
I don't think it's too much to assume, guessing by the name, that they have a strongly pro-American-in-all-things streak. Like the John Birch society I'm getting the impression that America is placed over and above the Catholic Church there, and that probably accounts for the whiff of Masonry. Vatican II, after all, is nothing but the Americanized Masonic anti-church. America has always followed the Masonic program, more or less unwittingly.
I know one member who became sede, he said his fellow Knights would not be amenable to his new ways at all.
Any Zionist influence there, pro-Israel sentiment? On the Internet I didn't find much that was grievously objectionable. The worst was a book review in one of their newsletters about Israel as a model business state. The article had lots of tea party rhetoric about how Israel is growing fast because it doesn't have governmental restrictions on business like the U.S. ( surely these people jest, where is the restriction on outsourcing jobs to Chinese slaves to make cheap Ipads and then marking them up to ridiculous levels back home? ). But it didn't sound religiously indifferent, it just sounded money-obsessed, like the writer saw everything through the lens of money and "progress." That is par for the course in America.