The freedom to vary $\lambda$ is severely constrained: For a nontrivial solution of the Killing spinor equation you need either a real and constant $\lambda$ or a function $\lambda$ with purely imaginary values [Lichnerowicz (1987)].
The generalization of the Killing spinor equation to nonconstant imaginary Killing function $\lambda$ has been studied in
H.-B. Rademacher, Generalized Killing spinors with imaginary Killing function and conformal Killing fields, Lecture Notes in Math. 1481 (Springer, Berlin, 1991).
See also Eigenvalues of the Dirac operator, Twistors and Killing Spinors on Riemannian Manifolds.