No. None of the organizations show titles of any other organization on the dog's pedigree. Breeders need to make up their own pedigrees in order to do that, and because some breeders either don't understand standard terminology, or want to play with it a little, you need to know what abbreviations mean and maybe ask to see the back up for such made up pedigrees. I am not saying people make the peds up to cheat, most make them up because it is the only way to get title from multiple organizations on one piece of paper, just be aware that sometimes there is fudging.
In fact, even AKC pedigrees do not show all AKC titles necessarily. For whatever reason, parent breed club titles are sometimes omitted from AKC pedigrees. The German Shorthaired Pointer Club of America ("GSPCA" is the parent breed club for GSP's under the sanction of the AKC. The GSPCA sponsors the National Field Trial and wins entitle dogs to a number of titles, such as "NFC" for National Field Champion. These titles are in the AKC rules. But for whatever reason, the titles awarded at the GSPCA Nationals sometimes do not show up. One sometimes has to contact the AKC and request that they be added.
As for your question about what event is most representative of hunting, I can give you my opinion. That event would be any of the wild bird field trials that happen under the American Field umbrella, and the most representative of all would be what are called Cover Dog or Grouse Dog trials, because those are not only run entirely on wild birds, but are also entirely foot handled, which is how the vast majority of hunters work their dogs. The other AF wild bird trials are horseback handled. I have horses, and have ridden and participated in many horseback trials. I love the prairie grouse (sharptail) trials to death, but I think the ruffed grouse trials are a more realistic representation of hunting.
All hunt tests, all AKC sanctioned field trials, all NAVHDA tests, and most field trials outsides of the AKC umbrella are run on planted birds these days, and you can argue that they are representative, but no wild bird smells of bird crate, human hands and bird poop. And no wild bird will sit there while you pick it up and throw it. Well, I saw one wild rooster in the snow that did that once, but for all intents and purposes, it never happens.
In GSP circles, we have only a handful of wild bird trials, all horseback and all under the NGSPA, they are the Sharptail Championships, the Greater Prairie Chicken Championships, and the Region 17 Championships, which are run on wild birds. On occasion, wild birds are encountered in the hour Championships. Very rarely in the AKC half hour stakes.
All the other trials and hunt tests are actually more representative of the training dogs undergo, they are tests of the dog's ability to perform as trained.