You store it as an integer, but as we just saw in the OP, for general interop with any system that parses JSON you have to assume that it will be parsed as a double. So to avoid precision loss you are going to have to store it as a string anyway. At that point it's upto you whether you want to reinvent the wheel and implement all the required arithmetic operations for your new fixed-point type. Or you could just use the existing decimal type that ships on almost every mature platform: Java, C#, Python, Ruby, etc.
In dollars, what do you get up to with a double of cents without precision loss? It's in the trillions, I figure? So a very large space of applications where smallest-denomination-as-JSON-number is going to be fine.