This is a simple one to answer. Define what criteria make a 'big club' and then decide if City meet those parameters.
If I had to define the criteria it would simply come down to regularly winning trophies in a country where the football is recognised as being up there with the best in the world.
You could perhaps make a case from a financial standpoint, and then the likes of the Rags come into play, but by my criteria they are nothing more than a global brand that generates a vast amount of revenue. When we play them, or the Dippers play them, they get ripped to pieces. Financially they are big, performance wise they are little more than a footnote.