Your problem is rather complex and a bit specific, I will try to answer outlining general steps.
These steps will seem cumbersome but are aimed to answer your question even if they stray away from it at some point. But I am outlining them to help you define an strategy.
First, calm down and doubt yourself
You can't even start deciding on a plan of action until you have calmed down.
It is obvious this problem is causing you and your co-trainers distress.
Why doubt yourself? If you can't think at least for a moment "I might be wrong", then you haven't calmed down. There is always a possibility that you could be wrong, even when it doesn't seem like it.
Identify and write down the facts
You need to separate what you know from what you feel. But what you feel is a fact also. Just focus on your feelings about yourself and the situation, not about the other person.
- There is a performance group
- There is a freetime group
- The groups don't have clearly defined purposes
- You and all the trainers care for the kids and for the groups.
- For you and other 2 trainers solving the problems of your group is more important than your relationship with the trainers
- For the other trainer the relationship is more important.
- The relationship is important for everyone.
- For the Club board the performance group is priority over a bunch of kids "hanging out" (which apparently is their opinion of the free time groups).
- You need to solve this before the end of December.
I just stated what seem to be facts, it doesn't mean anyone in particular is right.
What is the real problem? / List of issues
You need to identify all the issues in a situation in an objective way to start working things out.
Consider everything that is not clear or in a grey area.
Consider anything that is causing conflict, for example the fact that your relationship with the other trainer is not good.
You said that the problem is that someone in your group is not being objective and doesn't want to show up in your meeting.
From your description, that doesn't really seem to be the problem, or at least not the only one.
However, you haven't described what are the personal issues going on, or the group issues either. I assume they are of sensitive nature and that is why you avoided going into details.
I will list general cases and what to do in each case, starting from your question.
You said he is not being objective, but that depends. You are 3 against 1, and just looking at that fact, you are right. But remember, democracy is not a perfect system, there are cases in which the minority is right, you can look into history for examples.
Not life changing decisions: In many cases if the minority doesn't go with the majority vote, they are wrong. This happens a lot with decisions that are not life changing, but someone insist on getting his way, over the other's.
Note that the minority may actually be right in these cases, but are wrong on trying to impose their opinion over the majority's.
In this case, if it is so important for him, and it is not so important for the majority, it might be healthier to let the person get his way. Is it something that will affect you for a long time or is it something that will last for a limited time?
If it is something important for the majority also then their decision rules over the minority's.
Meaningful issues: These are moral issues, things that can change someone's life forever or for a significant amount of time. Things that can hurt a third party, etc..
From the description of your issue, it falls in this category. You are talking about things that affects a lot of kids, therefore you want to be careful. If it would only affect you and the other 3 guys, then it would be fine to rule with the majority vote and to get rid of the trainer that doesn't want to cooperate.
You have to think that the he might have in mind the kids well-being when he refuses to cooperate.
When you are handling kids the relationship between the trainers/tutors/teachers is very important. You are the role models and people that they have to be with a lot of time.
Think of it as a dysfunctional married couple. They may try to do the best and be civil, but the children still feel the issues, and it can be more damaging than if the parents got divorced.
What are these "relationship issues" you are talking about? Do you guys hate each other because of personal likes? Is he a rude person? Does he insult you or belittle others? Doesn't he listen to what others have to say? Basically is he someone you don't like working with? Or the issue merely come from work disagreements? From your description it seems the latest.
What are the "group problems" you want to solve? Are they administrative issues? Like schedule, materials required, etc? Or are problems about the kids? Are they about the purpose of the performance and free times groups and training schedule? If they are administrative issues, they can wait, the work disagreements are more important. If the group problems you describe are basically how to carry on with the groups the next year, well that is the existing work disagreement you have with the other guy.
In this case you can tell him that is what you are going to be talking about.
What is the root cause of your problem?
You need to identify the real reason of your issues. What you are seeing is the symptoms of a major issue. Many times this is the case.
You are trying to go around the fact that the mission of both groups are not clearly defined, and your board seems to have a bad preconceived idea of groups that are not "performance groups".
Is your club a non-lucrative organization? Or are they trying to make money out of the groups? In both cases, shouldn't they want the biggest amount of kids to participate?
Sport is way more than just top performers.
If your company only wants to focus on performance groups, they should take that decision, but they need to realize most people are not into "top performance" and other clubs will get a lot of the kids away from them.
If however, as it seems from your description, they like the "performance" thing just because it presents the club better. They just need to promote the groups in a smart way. It is not that "performance group" presents the club better, it is that they are not looking at it correctly.
Sports are very important for kids, not only in high competition levels, but because it helps mental and physical health, fosters group work and other positive social characteristics. Well you probably know more about it than me.
You can separate the groups and still have a commercially attractive image. You just need to advertise things properly.
If however you can not separate the groups because of logistical reasons, you could logically separate them. Have the free time group focus in a more casual training schedule. That can be good for the performers also, since you can not make them train so hard all the time and will keep the fun factor up.
And you could even have additional exercises for the kids of the performers group during the free time group.
It seems to me that the other trainer doesn't think a free time group should train so harshly, and he is right. You can not have a so called free time group train like a performance group, it will discourage kids and make them feel bad. And it is also a sort of false advertisement.
There is not enough time
This rarely is true, normally is more about wanting to make the time. Sacrificing other things, etc.
Performance is important, but it is not the main thing. Out of 100 kids maybe 10 can be top performers? Even if 50 kids out of 100 can be top performers, is it OK to ignore the other 50?
You have a freetime group which is where many kids that are not into top performance will like to join. If you make them train like the performance group you will discourage them and they will likely leave, hopefully they will keep practicing the sport in another place, but they might not.
Resources and deadline
List all the resources you have. How much are you willing to work on this, your time and energy are resources also, and if you wear yourself out is not good.
How much time you can delay the decision, etc..
Take a decision
Having outlined all the issues you can decide which ones you want to tackle down. Consider the resources you have and what can be done in the short and in the long time.
Only you know all the background, so only you can take this decision.
Also, consider what kind of example do you want to set for the kids. Is it ok to kick a person because you don't agree with them or is it better to try and work problems out?
Once you have done this, you can approach him.
If you truly think there is no way you can solve the relationship issues within this month and that it is imperative to solve the group problems, then approach him in private. Be empathic, let him know you are aware of the other issues and you really want to solve them. Explain him that it will take time, and you need to solve the group issues urgently.
Propose a date to start working on the other issues, and make sure it happens. Ask him to make a proposal.
Make him see that if the 3 of you don't agree the only ones affected are the kids.
If there are some "demands" he has, try to concede some of them.
He probably also agrees that you are short in time. Start working the conversation from the points you agree in.
For me it is obvious that you want to keep both groups and that you value the free time group also. Make him see that.
Finally, one of the sources of this conflict is the Club board point of view and the resources you have (trainers, club facilities, etc). Make him see that you need to work on changing the board point of view towards the free time groups. And that you need to work on a detailed schedule to make both groups work, time you simply don't have now.
Things haven't gotten this way all of the sudden, probably it has been like this for a while, so they can remain the same for a while longer, as long as you really intend to fix them and you have a plan for it. He probably can agree on this also.