You're right, the Standard of Ur would not work. It is far too busy.
But....
What if we took the three main colors from the Standard of Ur and used them on the tricolor? Suitably muted, of course, to blend with the overall image.
It looks like the main colors of the Standard of Ur are a rather bright blue, dark red and gold.
Other ideas
My first thought of a team graphic was the flag of the Duchy of Grand Fenwick, from the movie 'The Mouse That Roared'. To me, the 'Federal Republic of the Evaacck Empire' has a pompous, medieval sound to it. So I thought of heraldic images of that time.
Team Image Idea 1
I thought we might be able to change the words from 'Yay' and 'Nay' to 'Team' and 'FREE'.
When aluka mentioned "Liberty Leading the People" I thought of Mel Gibson in The Patriot.
Team Image Idea 2
Good picture, but seemed a bit hard to modify for our uses.
Team Image Idea 3
Awesome photo, sacred to the US Marine Corps, but maybe a bit heavy for what we want.
Anyway, these were my ideas, but they are only starting points.
I think aluka's idea is better than any of mine.