Missouri has state records for "other methods" which includes snagging and bowfishing. So it counts for that. As far as I know, there is no group that maintains official or unofficial records of USA or North American fish, so there is not anything to qualify for there. IGFA won't accept snagged fish, so it is not eligable for a world record. There was one bighead carp over 150 pounds caught in Europe two years ago, and at least one of over 135 pounds in Brazil since then, but neither of these were submitted for a world record, and probably would not have been approved due to gear type and how the fish were handled. If you want an IGFA record, you have to go prepared to get it. Both of those big fish were eaten by people hungrier than you and me and less concerned about holding any kind of record than they are in a good meal.
BAA keeps its own records, of course, and I think Darrin Opel's 92.5 pound fish is still the bowfishing record. Unfortunately, it is not the Illinois record, because it was caught before Illinois records on bighead carp began. You'd think they would want to own this record, but have not. Tennessee has a state record bighead carp of a couple ounces bigger than Darrin's, according to the records caught "by hand". Crazy. I don't know how you catch a large bighead by hand unless it is dead or dying.
The largest silver carp I know of in the USA was 74 pounds, caught by a commercial fisher, and it is hanging on a wall in Spirit Lake, IA. There are records of silver carp approaching a hundred pounds in Asia, again, caught by commercial fishers. The largest silver carp I have held in my hands was only 24 pounds, and I have been fishing them hard for ten years. Fish larger than 20 pounds are uncommon, although every time someone gets hit in the head by one they report it was a twenty pounder. People usually overestimate the weight of silver carp. That said, I'm sure there are some that frequent this board that have taken larger fish than I have. My gear is not designed to take the really large ones, and I have seen bigger fish get away.