First I am going to say that I do not know you your horse or the people at the shows. You say local rodeo?? Not sure if you are actually talking a RODEO or just a local show with several events.
First if it is a local show with several events and you are there to have fun they enter other events and skip the speed events. This will help your horse to be well rounded and you will not be competing against these people. This is what I do. When a local show offers reining I do not enter the reining class unless I have a green reining who I am schooling and I enter the class with the understanding I am JUST Schooling and not to judge the horse. Just would not be fair any other way in most cases.
If there is more then one level of speed events like there is around here even at the local speed shows as they are NBHA move up in class.
Other then that just suck it up or do not go to the small shows just the larger ones where you have more competition.
Also the way YOU act when you win is just as important as how you act when you loose. Might want to think about that too.