I don't agree with the try outs for a team, it should be open to everyone. But A good idea could be to use a point system, based on attendance and just general shown interest in the team. All you would need is a chart where every student has to mark/sign in at each meeting, students who don't show up much, are considered a non traveling member, unless they pay all for them selves. And i understand there are always odd cecum stances that could make people miss time and such but those could be dealt with on a case by case basis.
Having a application is a great Idea, MOE uses one since we are part of 12 schools we have to have a way to sort people out. Here is MOE's student
application I hope the best for your team and for you to be able to come together as a team, so you can continue to be a FIRST team.