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We consider massive half-integer higher spin fields coupled to an external constant electromagnetic field in flat space of an arbitrary dimension and construct a gauge invariant Lagrangian in the linear approximation in the external field. A procedure for finding the gauge-invariant Lagrangians is based on the BRST construction where no off-shell constraints on the fields and on the gauge parameters are imposed from the very beginning. As an example of the general procedure, we derive a gauge invariant Lagrangian for a massive fermionic field with spin 3/2 which contains a set of auxiliary fields and gauge symmetries.
Buchbinder, I. L., Krykhtin, V. A., & Tsulaia, M. (2015). Lagrangian formulation of massive fermionic higher spin fields on a constant electromagnetic background. Nuclear Physics B, 896, 1–18. https://doi.org/10.1016/j.nuclphysb.2015.04.008