Does a Berry phase operator exist?

There is no single operator corresponding to the Berry phase because the Berry phase is not a property of a single quantum state, but of a full trajectory through state space. "Observables are Hermitian operators" does not mean "anything you can measure is a Hermitian operator", it means measurable properties of states are Hermitian operators acting on those states.

It easy to construct plenty of other things you can "observe" that are not actually observables/Hermitian operators, e.g. if you do repeated measurements on an identically realized state, you can compute (an approximation for) the standard deviation of the observable being measured, but the standard deviation itself is not a linear operator on states, either.