There isn't such a thing as that in base, but well done for asking the most entertaining question for me on Stack Overflow for some time. Functors and Applicative functors are closed under composition (which certainly isn't the case in general for monads, thus the need for monad transformers), which...

haskell,monad-transformers,lifting

You have two options: Find a monad morphism. This is often a matter of finding the right library; in this case hoist and generalize together should get you where you need to go. Make your State action more polymorphic. This is the commonly used one, and the recommended one; it...