How can I say whether a Hamiltonian is integrable or not?
I don't think that level spacing is "enough" to determine a system is "integrable" or not. (of course it depends on how one defines integrability.) The level spacing idea is called Berry-Tabor conjecture, and it is not proven that Poissonian distribution is intrinsic in the case of quantum integrability.
To me, the existence of extensively many conserved charges (with local or quasi-local densities) suffices the "quantum integrability". (or equivalently, the existence of Yang-Baxter equation in the system) Many systems like Lieb-Liniger model and Heisenberg XXZ chain are solved by Bethe Ansatz, while some others are solved using Yangian symmetry, e.g. long-range Haldane-Shastry model.
Of course, if a model after some transformation becomes a free model, as in the case of transverse Ising model, it is integrable. (scattering in free model is trivial and infinitely many conserved charges with local densities are easy to construct.) In general, there is no a priori way to determine whether an interacting quantum system is "integrable" or not.