Stopping Others From Using the Name. You might be able to register the trademark PROJECT IRON QUILT as a service mark since there are no substantially similar marks. Possibly the owners of the IRON CHEF mark may object (although we doubt it). The owners of the PROJECT RUNWAY mark may care more about it since quilts are closer to fashion than cooking. But even if you obtain the mark, that doesn't guarantee your use of the concept.
Staying below the radar is better. Under various copyright (you're creating a derivative version) and unfair competition principles, the owners of the game shows may object to the borrowing of their formats for your purposes. (Reality shows are protective of their respective turfs.) Neither show may want to intrude on you if you're just running a charity benefit and raising money for cancer patients. But once you begin to claim IP real estate by registering the name or seeking other proprietary rights the shows may feel they have to react to preserve their property. That's why the Dear Rich Staff recommends you stay below the radar and enjoy your success without the paperwork.