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State-space models (SSMs) recover low-dimensional latent dynamics from high-dimensional observations, but latents are generally not identifiable from observational data alone. Interventional state-space models (iSSMs) use perturbations to break observational symmetries and identify parameters up to permutation, scaling, and shift. However, prior work restricts identifiability to linear latent dynamics and piecewise-linear emissions, excluding many behaviors characteristic of neural circuits. We extend iSSMs to a substantially broader model class: we prove that (i) affine identifiability holds under general injective emission functions and Gaussian latent transitions, and (ii) with interventions, identifiability up to permutation, scaling, and shift is preserved under general nonlinear latent dynamics. We validate our theory on simulated FitzHugh-Nagumo dynamics, and then demonstrate two applications: adjudicating between competing models of decision-making; and predicting responses to unseen micro-stimulation in macaque and human neural recordings, with proof-of-concept generalization to a held-out stimulation site in the latter. Finally, simulations yield practical guidance for designing stimulation protocols that improve identifiability.