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