Why is finding the ground state of a Hamiltonian in QMA? Why is finding the ground stte of a Hamiltonian in QMA?

It's in QMA to figure out if a hamiltonian has any energy eigenvalue within a certain window range which is at least inverse polynomial in size. But what if the energy gap for the ground state is inverse exponentially small? Then why is specifying the ground state in QMA? It takes an exponential time to measure energy up to inverse exponential accuracy, right? So, Arthur needs more than polynomial time to verify, right?
There needs to be a gap between the eigenvalue of the groundspace and the first excited state that is inverse-polynomial in the system size, and indeed such a promise is required for the QMA-complete Local Hamiltonian problem. Please refer to the formal definition of the Local Hamiltonian problem and the QMA-completness proof, e.g. Definition 10 and Theorem 1 in Nagaj's thesis. If the gap is inverse-exponentially small, then the promise is not satisfied by that local Hamiltonian instance and the QMA-completness proof does not apply since a key assumption is not satisfied.