Dizzyingly eclectic means that the food comes from amazingly many sources. The fare on offer is the food that is offered to people in the restaurants. A bill of fare is the same as a menu, but much less common.
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tinanam0102on offer" means on sale at a reduced price, but I don't think it fits here.You are right about on offer. It may or may not have exactly that meaning here. I think it's more likely that it simply refers to the various dishes and kinds of food offered to customers.