If only this could be appealed...
Poll taxes were outlawed (other than buying an ID, which can be gotten free if you're poor enough), we wouldn't dream of the gov't charging people to go to church/temple/mosque, so why is it OK to tax & license this civil right?
It could be appealed, but you'd have to start over again. If your basis is going to be constitutional, why not make it a federal case instead of asking the state to pay it bacK? A loss, if there is one, could be appealed just like any other case. But, a generalized "I shouldn't have to pay a fee" claim is a non-started. Fundamental constitutional rights are licensed and "taxed" regularly. There is a constitutional right to marry, but every state requires a license and charges a fee. Same for driver's licenses (unless you drank the driver's ed coolade that driving is a privilege, which is the topic of another thread.
In order to have a chance at a case, you'd have to be a plaintiff that cannot afford a license, or a plaintiff that for some strange set of circumstances has to pay a lot of money. Such a plaintiff might have a chance.