Here's the story of someone in the same boat. When he bought his house, the previous owner added on an extension -- without a permit
When it was time for him to sell, buyers wanted the extension to be permitted. He even got into contract. But the process of getting a contractor to design the plan, getting the city permit department to approve the plan, and getting the work done took longer than he wanted, and the buyers backed out.
Other buyers didn't want to buy a house with unpermitted work, especially since this was already known to the city.
So the seller took time off, got the permits, got the extension done correctly (ripped out the old, put in new according to code), submitted request to have the additional 240 sq ft recognized and had the public records changed.
When done, he put the property back on sale (minus some $$ for the cost of permits and construction), but also sold the property quickly and at over his last list price.
If you choose to fight the city's assessment -- you may be asked to
1) remove the extension since it was done without permits,
2) get a permit and pay fines for the extension, and have the work done according to code
And if you choose #2, when all is done, go back to county records and have them reflect the correct square footage on the public records. When it comes time to sell, you have the peace of mind (and pride) that the extension is permitted and that the square footage is legitimate. It could be your competitive advantage over other houses that may come up for sale in your area but do not have the extra space.