It is possible to do something along these lines but you need some boilerplate unfortunately, so it would probably be better to go about it in a different way.
Here is one way it could be done though (using something equivalent to a dependent sum):
{-# LANGUAGE ExistentialQuantification, GADTs, DataKinds, TypeFamilies #-}
type family StringP a where
StringP String = 'True
StringP a = 'False
data CheckStringness a where
IsTypeString :: CheckStringness String
NotTypeString :: (StringP a) ~ 'False => CheckStringness a
data Showable = forall a. Show a => Showable (CheckStringness a) a
instance Show Showable where
show (Showable IsTypeString str ) = str
show (Showable NotTypeString other) = show other
The difficult part is that you cannot directly reflect a type into a value in the way that you would want to for this, so you have to write a bit of boilerplate code to take care of that.
Example usage:
ghci> show (Showable NotTypeString (123 :: Int))
"123"
ghci> show (Showable NotTypeString ())
"()"
ghci> show (Showable IsTypeString "abc")
"abc"
Like I said though, I would try to approach the problem in a different way entirely (such as Luis Casillas's and ErikR's recommendations in the comments on this question), to avoid being in this situation in this first place. The main reason I demonstrated this is that things similar to this technique may at some point become nicer to work with and have more practical value than they do now, especially as the dependent Haskell initiative continues.