There should be an apostrophe to make "plate's" possessive. "Bagel" is acting as an adjective. The "base" part of this food is a bagel, with other things on top, so the food has a "bagel base." (This isn't a common expression or anything, but it makes sense in context.) The word "plate," like "dish," is being used to mean "a particular type of prepared food," and "hangs in the balance" is an idiom meaning that the fate or outcome of something is uncertain. It sounds like the speaker thinks that using bagels as the base for the dish may have been risky: the main part of the dish (the bagels) may or may not impress the judges (or whoever is going to eat the food).