Fairs frequently operate on very different rubrics and precedents than ARBA shows. Very often they are run and judged by volunteers (many ARBA judges expect a stipend, which many fairs don't have the budget for); those volunteers may or may not be up-to-speed on ARBA standards or show rules, and in my experience they put in long, sometimes thanksless hours of effort. In my experience, fairs are also hesitant to disqualify animals because owners (who also often do now know about or understand the ARBA SOP) provide the exhibits that make up a big part of the fair's bread-and-butter, and so the fair organizers justifiably do not want to alienate them.
Part of rabbit showing is the sportsmanship you mentioned teaching your daughter. It is considered poor form to correct a judge, especially in public, but there is nothing stopping you from having a discussion with her about a breed standard of perfection. A person only knows what they know, and don't necessarily know things may have changed. Mini Rex were only accepted in 1988, which is not that long ago for some of us! You might be in a position to open up a whole new world to her. I am very strongly supportive of education of fair organizers, breeders and students in regard to the standards governing the various breeds of livestock.
Perhaps you could offer to provide the fair with a scale and ARBA SOP. If you know and understand the SOP, you might offer to hold a clinic on reading the SOP, general DQs and judging techniques. That's what we did for our state fair several years ago, and the quality of both rabbits and judging has increased tremendously. In fact, for the last several years, we've run numerous rabbit and poultry clinics for 4-H students and leaders, to help get everyone up to speed on reading, interpreting and breeding to a standard. It's a foreign concept to many people, and quite different from the standards they may have encountered elsewhere, like those for dogs, horses, or other livestock.
IMO, I don't think an ARBA-licensed judge is necessarily critical; successful breeders, ARBA registrars, or those in training can often do a great job of judging, which is usually how our fair rabbit show is judged. As I said, many fairs have a "kinder-and-gentler" approach to judging and decline to disqualify an animal for anything other than the wrong sex or an obvious health issue, and that practice can put some ARBA judges in an uncomfortable spot.
Incidentally, the head and ears on a Mini Rex are only worth 5 points each, and as long as the ears aren't over 3.5 inches, there's no DQ, even in the ARBA SOP. A compact head, ears and body really says "Mini Rex,"and you are correct that the body should be compact, but lacking that is only a fault; the only type DQ in the M.R. SOP is for "extremely low shoulders, body which is extremely rangy or racy." That "extremely" means that it is up to the judge's discretion, which at an ARBA show, at least, is incontestable. Fur, color and condition are worth more than general type in the Mini Rex (55 vs 45 pts, respectively), so that could have factored into the decision. The weight issue is important in Mini Rex, but if the fair doesn't even have a scale, it sounds like the playing field is probably about level, with rabbits both under and over the min and max weights being included in all breeds(our fair does not outright DQ for weights, either, even in the meat pens).
It's hard to watch what you consider to be an inferior rabbit place over yours, but the fact is that no two judges will ever have quite same opinions; sometimes a judge will even reverse his or her own judgment on a different day. I've stood and watched my rabbit be DQd on one judging table, only to have it win Reserve in Show on another table. The human variation in judgement and SOP interpretation is a feature of rabbit shows that I actually appreciate, since it gives more people a chance to win.
