= Solution
The spinor <basis> in <3/c> is compatible with the supplied <gamma matrices>. The rotation generator $r(J)$ is anti-Hermitian and commutes with $\gamma_0=-i\sigma_1$. The two boost generators are <Hermitian matrices> and anticommute with $\gamma_0$. It follows separately for all three generators, and hence for every real $T\in\mathfrak{so}(1,2)$, that
$$
\boxed{r(T)^\dagger\gamma_0+\gamma_0r(T)=0.}
$$
For the infinitesimal <Lorentz transformation>, $\delta\psi=r(T)\psi$ and $\delta\psi^\dagger=\psi^\dagger r(T)^\dagger$. Hence the <Dirac adjoint> transforms contragrediently:
$$
\delta\overline\psi=\psi^\dagger r(T)^\dagger\gamma_0
=-\psi^\dagger\gamma_0r(T)=-\overline\psi\,r(T).
$$
The two contributions to the scalar variation cancel:
$$
\boxed{\delta(\overline\psi\psi)
=(\delta\overline\psi)\psi+\overline\psi\,\delta\psi=0.}
$$
This proves the <spinor invariant in three spacetime dimensions> has infinitesimal <Lorentz invariance>. Real transformation parameters matter because the <Hermitian adjoint> conjugates coefficients; this is not an invariance under arbitrary complexified parameters. For ordinary commuting spinor components, the bilinear is purely imaginary in this mostly-plus convention, and multiplying it by $i$ gives a real invariant. Its reality is separate from its invariance.
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