The prolongation structure method developed by Wahlquist and Estabrook is one  method to show whether or not a PDE is solvable via the inverse scattering transform. (There are others - refer Y. Kosmann-Schwarzbach, B. Grammaticos, K. M. Tamizhmani (eds.), *Integrability of Nonlinear Systems*, Lect. Notes Phys. 638 (Springer-Verlag 2004.)

For example, in an appendix to their second paper (on the NLS equation - J. Math. Phys. 17 (1976) 1293-1297) Estabrook and Wahlquist analyse the KdV-like equation $u_t + u_{xxx} + f(u)\,u_x = 0$, concluding that it admits a non-trivial prolongation structure (i.e. is integrable via IST) only if $f(u) = 2\alpha + 6\beta u+12\gamma u^2$, for some constants $\alpha$, $\beta$, $\gamma$.

Dodd and Fordy *"The prolongation structures of quasi-polynomial flows"* Proc Roy. Soc. Lond. A**385** (1983) 389-429 discuss methods for dealing with a general class of equations that includes your variants of the NLS.  Applying their methods to your quoted example would show why it does not produce a non-trivial prolongation structure.

As far as I can see from the literature, perturbations of the type you describe are discussed as perturbations of the IST solution of the unperturbed equation, e.g. V. I. Karpman *"Soliton Evolution in the Presence of Perturbation"* Physica Scripta 20 (1979) 462