Look, you guys, the answer is right in front of you. Triangles, circles, in a square, and Trinity Church. What one object is a combination of all of the first three elements, and also shows each individually (like the Trinity, for Christians, almost)?
A TRAFFIC CONE!
I think the game piece is a traffic cone. Lay out an obstacle course. Bonus points for beating it in the end-of game automode.

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Past teams:
2003-2007: FRC0330 BeachBots
2008: FRC1135 Shmoebotics
2012: FRC4046 Schroedinger's Dragons
"Rockets are tricky..."--Elon Musk
