If I'm understanding the team trophy is given to the studio. I think it's okay if you keep them or at the very least keep the special ones just to keep the peace.
Here's my idea for distribution. Have a "drawing". Give each dancer 'tickets' to put their names and phone numbers. I think the number of tickets could be based on how long they danced with you or how many competitions. Maybe one ticket per paid month of dance. You can just cut up slips of paper.
In front of each trophy goes a bag labeled with the trophy information and dancers puts their tickets into the bags. If they especially desire a certain trophy they put more tickets in that bag.
I would also put out a "I don't want a trophy" charity bag. If someone doesn't want a trophy they put tickets in this bag and you donate 25 or 10 cents per ticket to a specified charity. This just makes it more interesting for those who don't want a trophy and also might encourage people to think 'hey, we don't really need a trophy'.
To avoid hurt feelings and drama you hold the "drawing" privately and deliver or have people pick up trophies at a later date.
Now here's why I keep putting "drawing" in quotes and why you hold the "drawing" privately...if you need to you can fix the "drawing". No one pays for tickets and you never promise that it will be random. What you're really doing is seeing what everyone really wants. Even privately, you could draw random names but this at least gives you the chance to see that Susy only did team dances and put all her tickets in one bag and give it to her. This also keeps someone from randomly getting several.
Hope this makes sense and is helpful!