Sure they did:
Not a single team member has profited from winnings.
Each of those things are profiting from winnings. In the first two there is also a tax deduction, and, in the latter, a direct benefit.
The winnings have been split up between tithing to our church, giving half of the remainder to our churches small groups fund, then using the balance to further our ventures on the circuit.
Is this worth me pursuing for non-profit tax exempt?
You have a schedule C. You earn money. You deduct what it costs to earn the money. (What you spend to "further your ventures".) That will leave some left. You can donate it to valid charities without receiving anything and it will be like it was not income. The only issue will be the SE tax (SS and medicare) on the earnings above a certain amount. (Maybe some income limit issues, depending on your whole tax situation, as well.)
Odds are it really isn't going to make a whole lot of difference how you organize. Compliance costs will probably exceed tax savings. But, how much are we talking about?
When you are a Bear of Very Little Brain, and you Think of Things, you find sometimes that a Thing which seemed very Thingish inside you is quite different when it gets out into the open and has other people looking at it.
--W. T. Pooh (aka A. A. Milne)