You could have asked the same question about a spin one field. Why do they transform in the $(\tfrac 1 2, \tfrac 1 2)$ representation and not in $(1,0) \oplus (0,1)$? The reason is gauge invariance; the gauge fields $A_\mu$ transform in $(\tfrac 1 2, \tfrac 1 2)$, but the gauge invariant field strength $F_{\mu \nu}$ transforms in $(1,0) \oplus (0,1)$.

The same holds for the gavitino. The Rarita-Schwinger field $\psi_{\mu \alpha}$ is like the gauge field $A_\mu$. It has a gauge transformation $\delta \psi_{\mu \alpha} = \partial_\mu \chi_\alpha$. Its gauge invariant field strength $\partial_\mu \psi_{\nu \alpha} - \partial_\nu \psi_{\mu \alpha}$ transforms as $(\tfrac 3 2, 0) \oplus (0, \tfrac 3 2)$.

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