I wouldn't think the organization would have to be ancillary. It could be it's own organization, and just co-sponsor the fleadh. But that would be at least a year or two down the line, from an organizational standpoint. Although I have to say, I don't think a serious argument can be made that anyone and everyone with an interest in learning Irish traditional music and in competing in the fleadh, is being excluded. There is no ethos or rules that says one must be a Catholic to participate. That is why the organization has it's charter non-denominational. So the Protestants feel free to participate. And obviously enough do, because down through the years, a number of them have been participants and winners in the competitions.
All that said, regardless of what you personally choose to do in future about the Irish Catholic ethos of the organization, you still are in the dilemma over this year's event. But that is your dilemma, not the organizations. I maintain it would be very wrong to force the issue at this year's event, by refusing to schedule a mass, or scheduling an ecumenical service in it's place. You might be able to schedule an ecumenical service in addition to a mass, but even that will likely leave a bad taste for some members.
Finally, my question is, how fair are you being to to the immigrant community that has been sponsoring this event for decades, by demanding they cleanse themselves of their Catholic identity to be "fair and inclusive" of others? Like I said, I can't imagine anyone suggesting that secular klezmer music organizations ought to be cleansed of their Jewish ethos to make the organization more appealing to Christians. Pretty offensive, I would say.