= Solution
Use the mostly-plus <Clifford algebra> and <Riemann curvature tensor> conventions of the preceding solutions, with $e=\det(e_\mu{}^a)$, $\kappa^2=8\pi G$, and a Majorana <gravitino>. An <action> for <minimal four-dimensional supergravity> in the exact transformation normalization given is
$$
\boxed{S_0=\int d^4x\,e\left[\frac{R}{2\kappa^2}-2\bar\psi_\mu\gamma^{\mu\nu\rho}\nabla_\nu\psi_\rho\right]+O(\psi^4).}
$$
The kinetic coefficient is tied to the normalization of the <supersymmetry> transformations: if $\Psi_\mu=2\psi_\mu$ and $\eta=2\epsilon$, this becomes the canonical coefficient $-1/2$, with $\delta e_\mu{}^a=(\kappa/2)\bar\eta\gamma^a\Psi_\mu$ and $\delta\Psi_\mu=\nabla_\mu\eta/\kappa$. Mixing those two normalizations would spoil the cancellation.
To the retained order, use the torsion-free connection. More generally the <1.5-order formalism> allows its variation to be omitted after imposing its own algebraic equation; the resulting contorsion is quadratic in <fermions> and first affects the displayed <action> at quartic order. The <metric tensor> variation induced by the <vierbein> is $\delta g_{\mu\nu}=4\kappa\bar\epsilon\gamma_{(\mu}\psi_{\nu)}$, so the <Einstein-Hilbert action> varies as
$$
\delta S_{\rm EH}=-\frac2\kappa\int e\,G^{\mu\nu}\bar\epsilon\gamma_\mu\psi_\nu+\text{boundary}.
$$
For the fermionic <action>, varying both Majorana factors gives the same contribution after <integration by parts> and the Grassmann bilinear interchange rule. Thus
$$
\delta S_{\rm RS}=-\frac4\kappa\int e\,(\nabla_\mu\bar\epsilon)\gamma^{\mu\nu\rho}\nabla_\nu\psi_\rho
=\frac4\kappa\int e\,\bar\epsilon\gamma^{\mu\nu\rho}\nabla_\mu\nabla_\nu\psi_\rho+\text{boundary}.
$$
The vector-index <Riemann curvature tensor> contribution vanishes by the <first Bianchi identity>. The spin-index contribution is determined by
$$
\gamma^{\rho\mu\nu}\nabla_\mu\nabla_\nu\zeta
=\frac18\gamma^{\rho\mu\nu}R_{\mu\nu ab}\gamma^{ab}\zeta
=\frac12G^{\rho\sigma}\gamma_\sigma\zeta.
$$
For the last equality, expand the antisymmetric three-gamma/two-gamma product: the totally antisymmetric <Riemann curvature tensor> terms vanish by Bianchi, and the surviving terms are the Ricci contraction minus half its trace. Hence
$$
\delta S_{\rm RS}=\frac2\kappa\int e\,G^{\mu\nu}\bar\epsilon\gamma_\mu\psi_\nu+\text{boundary},
$$
which cancels the gravitational variation. Variations of <vierbeins>, gamma matrices and the connection in the fermionic kinetic term carry three dynamical <fermions>. They lie beyond this linear-fermion cancellation and are cancelled by the omitted four-fermion completion. Thus the <action> is invariant up to boundary terms at the order specified, rather than claiming that a truncated <action> is exactly invariant at every <fermion> order.
For the cosmological extension, add a <gravitino> <mass> parameter $m$ and deform the derivative:
$$
\boxed{S_m=\int e\left[\frac{R+6m^2}{2\kappa^2}-2\bar\psi_\mu\gamma^{\mu\nu\rho}\nabla_\nu\psi_\rho
+2m\bar\psi_\mu\gamma^{\mu\nu}\psi_\nu\right],\quad
\delta\psi_\mu=\frac1\kappa\mathcal D_\mu\epsilon,\quad
\mathcal D_\mu=\nabla_\mu+\frac m2\gamma_\mu.}
$$
Indeed the <fermion> terms combine as $-2\bar\psi_\mu\gamma^{\mu\nu\rho}\mathcal D_\nu\psi_\rho$ because $\gamma^{\mu\nu\rho}\gamma_\nu=-2\gamma^{\mu\rho}$. The modified spinor connection curvature is
$$
[\mathcal D_\nu,\mathcal D_\rho]=\frac14R_{\nu\rho ab}\gamma^{ab}+\frac{m^2}{2}\gamma_{\nu\rho},
$$
and contraction gives
$$
\gamma^{\mu\nu\rho}\mathcal D_\nu\mathcal D_\rho\epsilon
=\frac12(G^{\mu\nu}-3m^2g^{\mu\nu})\gamma_\nu\epsilon.
$$
The same action-variation cancellation therefore works with $G$ replaced by $G+\Lambda g$ precisely for
$$
\boxed{\Lambda=-3m^2.}
$$
This is the <Anti-de Sitter deformation of minimal supergravity>. A cosmological term alone would leave its variation uncancelled; the <mass> term and the gamma shift of the <gravitino> transformation are both necessary. A positive <cosmological constant> cannot arise from this real unbroken minimal deformation.
For electromagnetism, retain $\mathcal N=1$ and introduce an Abelian <vector multiplet> $(A_\mu,\lambda)$, with Majorana <gaugino> $\lambda$. A complete quadratic-fermion extension is
$$
\boxed{\mathcal L_{\rm vec}=e\left[-\frac14F_{\mu\nu}F^{\mu\nu}-\frac12\bar\lambda\gamma^\mu\nabla_\mu\lambda
-\frac\kappa2\bar\psi_\mu\gamma^{\nu\rho}\gamma^\mu\lambda\,F_{\nu\rho}\right],\qquad F=dA.}
$$
The accompanying leading transformations are
$$
\boxed{\delta A_\mu=\bar\epsilon\gamma_\mu\lambda,\qquad
\delta\lambda=-\frac12\gamma^{\mu\nu}F_{\mu\nu}\epsilon.}
$$
They preserve the rigid Maxwell-gaugino <action> for constant $\epsilon$. For a local parameter, the variation contains
$$
\delta\mathcal L_{\rm rigid}=e(\nabla_\mu\bar\epsilon)j^\mu+\text{boundary},\qquad
j^\mu=\frac12F_{ab}\gamma^{ab}\gamma^\mu\lambda.
$$
To obtain this, vary $F^2$, vary both <gaugino> factors, and use $\gamma^{ab}\gamma^\mu=\gamma^{ab\mu}+g^{b\mu}\gamma^a-g^{a\mu}\gamma^b$. The terms with $\nabla\lambda$ cancel between the two <actions>; integrating the three-gamma term uses $\nabla_{[\mu}F_{\nu\rho]}=0$, leaving exactly the displayed derivative of the parameter. The interaction is $-\kappa e\bar\psi_\mu j^\mu$, whose variation under $\delta\bar\psi_\mu=\nabla_\mu\bar\epsilon/\kappa$ cancels that term.
The other linear-fermion variation is also fixed. Put $\mathcal F=F_{ab}\gamma^{ab}$ and $T_{\rm EM}^{\mu\nu}=F^{\mu\rho}F^\nu{}_{\rho}-g^{\mu\nu}F^2/4$. <Clifford multiplication> gives
$$
\mathcal F\gamma^\mu\mathcal F=8T_{\rm EM}^{\mu\nu}\gamma_\nu.
$$
Varying $\lambda$ in the gravitino-current interaction then gives $2\kappa e T_{\rm EM}^{\mu\nu}\bar\psi_\mu\gamma_\nu\epsilon=-2\kappa eT_{\rm EM}^{\mu\nu}\bar\epsilon\gamma_\nu\psi_\mu$, which cancels the Maxwell <vierbein> variation. This proves the required <supergravity coupling of an Abelian vector multiplet> through the retained order.
Beyond that order one uses a supercovariant <gauge field strength>, here $\widehat F_{\mu\nu}=F_{\mu\nu}-2\kappa\bar\psi_{[\mu}\gamma_{\nu]}\lambda$, and the associated higher-fermion transformation terms and four-fermion interactions. An auxiliary scalar may be retained for off-shell closure, or eliminated in this on-shell formulation. A <Maxwell field> without a fermionic partner cannot be coupled by merely adding $-F^2/4$ while leaving all transformations unchanged. An alternative is an $\mathcal N=2$ gravity multiplet, where the <vector> is a graviphoton and a second <gravitino> supplies its extra fermionic states; the explicit construction above is the minimal $\mathcal N=1$ matter extension.
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