= Solution
The conformal condition is a quantum <worldsheet Weyl anomaly> cancellation, not the classical metric equation of the <Polyakov action>. Assume a smooth target metric, no antisymmetric background field, and curvature/gradient scales large compared with $\sqrt{\alpha'}$. Use $[\nabla_a,\nabla_b]v^c=R^c{}_{dab}v^d$ and $R_{ab}=R^c{}_{acb}$, consistent with the Ricci-scalar <Weyl transformation> in the hint.
A sign convention in the printed action must be made explicit. With worldsheet signature $(-,+)$, continuation $\tau=-i\tau_E$ and $e^{iS_L}=e^{-S_E}$ sends its negative kinetic term to a positive Euclidean kinetic term, while its positive Lorentzian curvature coupling becomes a negative Euclidean curvature coupling. Define the conventional Euclidean <dilaton> by $\varphi=-\Phi$ for this literal action. Then $S_{E,\mathrm{dil}}=(4\pi)^{-1}\int\sqrt h\,R^{(2)}\varphi$. The <Lorentzian dilaton coupling sign convention> is therefore important: a convention with a negative Lorentzian curvature coupling would instead use $\varphi=\Phi$ and reverse every term linear in the printed $\Phi$ below.
For completeness, the metric part of the one-loop calculation can be obtained by a geodesic <background field expansion of a string sigma model>. Write $X=\exp_{\bar X}Y$. Its quadratic Euclidean action contains
$$
S_E^{(2)}=\frac1{4\pi\alpha'}\int\sqrt h\left(g_{ab}D_\mu Y^aD^\mu Y^b-R_{acbd}Y^aY^b\partial_\mu\bar X^c\partial^\mu\bar X^d\right).
$$
Here $D_\mu Y^a=\partial_\mu Y^a+\Gamma^a{}_{bc}\partial_\mu\bar X^bY^c$. In dimension $2-\varepsilon$, the ultraviolet coincident contraction has pole $\langle Y^aY^b\rangle_{\mathrm{div}}=\alpha'g^{ab}/\varepsilon$ with an infrared regulator. Contracting the curvature term gives a divergence $-(4\pi\varepsilon)^{-1}\int\sqrt h\,R_{cd}\partial\bar X^c\partial\bar X^d$. It is cancelled by the metric <counterterm> $\delta g_{ab}=\alpha'R_{ab}/\varepsilon$, giving the one-loop <sigma-model beta function> $\beta^g_{ab}=\alpha'R_{ab}+O(\alpha'^2)$ before the dilaton improvement. Equivalently its local Euclidean Weyl variation is $-(4\pi)^{-1}\int\sqrt h\,\omega R_{ab}\partial X^a\partial X^b$, with terms proportional to the embedding equations understood as field redefinitions.
Set $\Omega=e^\omega$ in the supplied curvature transformation. On a closed <string worldsheet>, <integration by parts> gives
$$
\delta_\omega S_{E,\mathrm{dil}}=-\frac1{2\pi}\int\sqrt h\,\varphi\Box_h\omega=-\frac1{2\pi}\int\sqrt h\,\omega\Box_h\varphi.
$$
The chain rule and leading <string embedding map> equation imply
$$
\Box_h\varphi(X)=\nabla_a\nabla_b\varphi\,h^{\mu\nu}\partial_\mu X^a\partial_\nu X^b+\partial_a\varphi\left(\Box_hX^a+\Gamma^a{}_{bc}\partial X^b\partial X^c\right).
$$
The kinetic embedding equation suffices for extracting the leading metric coefficient. More precisely, the full Euclidean embedding equation makes the parenthesis $(\alpha'/2)R^{(2)}\nabla^a\varphi$, giving a curvature contribution $-(\alpha'/4\pi)\int\sqrt h\,\omega R^{(2)}|\nabla\varphi|^2$ to the Weyl variation. This contributes to the scalar coefficient below rather than to the leading metric tensor coefficient. Combining the curvature <counterterm> anomaly with this variation gives the <leading metric-dilaton Weyl condition>:
$$
\boxed{\overline\beta^g_{ab}=\alpha'\bigl(R_{ab}+2\nabla_a\nabla_b\varphi\bigr)+O(\alpha'^2)=0.}
$$
This is the gravitational equation in the <string frame>, rather than an ordinary Einstein equation with a minimally coupled scalar. In terms of the field and signs literally printed in this Lorentzian action it reads
$$
\boxed{R_{ab}-2\nabla_a\nabla_b\Phi=0\quad\text{to leading order}.}
$$
The frequently used $R_{ab}+2\nabla_a\nabla_b\Phi=0$ is obtained if the printed $\Phi$ is identified with the conventional Euclidean dilaton, which requires the opposite Lorentzian curvature-coupling sign. Both conventions describe the same mathematics after $\Phi\mapsto-\Phi$, but one cannot change only the field equation silently. Also, vanishing of this tensor coefficient is the metric part of Weyl invariance; the dilaton curvature coefficient and central-charge condition remain to be checked.
Now derive the scalar consequence without presupposing its integration constant. The <contracted Bianchi identity> gives $\nabla^aR_{ab}=\tfrac12\nabla_bR$. The <Ricci identity> applied to the gradient of a scalar gives
$$
\nabla^a\nabla_a\nabla_b\varphi=\nabla_b\Box_g\varphi+R_{ba}\nabla^a\varphi.
$$
Diverging the metric equation and then substituting $R_{ab}=-2\nabla_a\nabla_b\varphi$ yields
$$
0=\frac12\nabla_bR+2\nabla_b\Box_g\varphi-4\nabla_b\nabla_a\varphi\nabla^a\varphi
=\frac12\nabla_b\bigl(R+4\Box_g\varphi-4|\nabla\varphi|^2\bigr).
$$
Thus the bracket is constant on each connected target component. Its trace equation is $R+2\Box_g\varphi=0$, so the <dilaton equation from contracted Bianchi identity> becomes
$$
\boxed{\Box_g\varphi-2|\nabla\varphi|^2=C,\qquad \Box_g\Phi+2|\nabla\Phi|^2=-C.}
$$
The second equation uses the literal printed sign convention. These are nonlinear scalar wave equations analogous to the <Klein-Gordon equation>. For the <exponentiated dilaton wave equation>, put $F=e^{-2\varphi}=e^{2\Phi}$; differentiating twice gives
$$
\boxed{(\Box_g+2C)F=0.}
$$
Here $\Box_g=\nabla^a\nabla_a$ is the Lorentzian <Laplace-Beltrami operator>, and $|\nabla\varphi|^2=g^{ab}\partial_a\varphi\partial_b\varphi$ is a Lorentzian contraction, not necessarily nonnegative.
The Bianchi and Ricci identities alone do not imply $C=0$. The <linear dilaton counterexample to zero integration constant> is flat target space with $\varphi=q_aX^a$: the tensor equation holds for every constant $q$, while $C=-2q^2$. A non-null $q$ disproves any deduction of the zero-constant scalar equation from that tensor equation alone.
The <leading dilaton Weyl anomaly coefficient> is obtained as follows. The matter and reparameterization-ghost <central charges> give the constant $(D-26)/6$. Expanding the curvature coupling along the same geodesic fluctuation gives the quadratic term $\tfrac12\nabla_a\nabla_b\varphi\,Y^aY^b$; its coincident contraction is cancelled by $\delta\varphi=-\alpha'\Box_g\varphi/(2\varepsilon)$, giving the term $-\alpha'\Box_g\varphi/2$. The full embedding-equation contribution identified above supplies $+\alpha'|\nabla\varphi|^2$. Therefore full leading-order bosonic-string Weyl invariance also requires
$$
\overline\beta^\varphi=\frac{D-26}{6}+\alpha'\left(-\frac12\Box_g\varphi+|\nabla\varphi|^2\right)+O(\alpha'^2)=0.
$$
It fixes $C=(D-26)/(3\alpha')$. Hence in the critical $D=26$ theory, or under <boundary conditions> making the constant vanish,
$$
\boxed{\Box_g\varphi-2|\nabla\varphi|^2=0,\qquad \Box_g\Phi+2|\nabla\Phi|^2=0,\qquad\Box_g e^{2\Phi}=0.}
$$
This explains precisely the extra input needed for a zero-mass Klein-Gordon-type equation. In noncritical dimension the corresponding equation has the central-charge-deficit constant, subject to the usual controlled-background assumptions.
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