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. A385 (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

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. A385 (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

Edit: most, if not all of the equations in two spatial dimensions - such as the Kadomtsev–Petviashvili equation - place heavy restrictions on soliton behaviour. They've sometimes been described as "one and a half dimensional" problems.