[mcouture]

There are some good ideas posted here already. Clearly you are describing a crisis scenario, one where neither "side" is willing to yield and listen to ideas from the other. Unfortunately, this is a result of leadership, or lack thereof. I don't intend to disparage whoever is at the head of your organization, but the buck does in fact stop there. Some leaders simply haven't thought much about the importance of leadership or considered what their strengths and weaknesses are and how that affects the team. The good news is that it is not impossible to overcome the problems you are all facing.

My suggestion is come together for (you guessed it) one more meeting. Only this time, it is with specific purpose and can be brief, but everyone must be there. You can refer to the meeting as a State of The Team meeting if you like. The purpose of the meeting is to accomplish something which both sides can likely agree on:
1) Recognize the crisis. Call it what it is. Agree that it is failing all parties.
2) Agree that it is absolutely necessary to find a constructive path to resolution.
3) Commit to allowing a third party to facilitate a series of meetings (as necessary) designed to map out new team guidelines that everyone can support and abide by.

It will be necessary that the third party is someone who is not part of the team, someone not associated with the students or mentors. This person will guide the process toward building a team that everyone can find reason to support and be excited to be a part of. This person may or may not have FIRST experience, but they will know how to conduct a meeting between the factions of your team. This goes way beyond simply spending more time together.

One of the outcomes of this will also be to define and identify the leadership needs of the team as well as the responsibility of leadership. This sometimes takes times, after all, poor leadership is what allowed the team to get where you are.

Understand that this will be a challenge to all parties and that there is a chance that an individual may discover he or she isn't going to be compatible with the new team and dismissal will be necessary. That's okay, your team will be stronger for it.