= Solution
Let $p(z)$ be the normalized galaxy <probability density> per unit <cosmological redshift>, $\int p(z)\,dz=1$, and let $b_g$ be its linear <galaxy bias>. In a spatially flat background with $c=1$, $d\chi=dz/H(z)$. Thus the projected <galaxy density contrast> is
$$
\delta_g(\hat{\mathbf n})=\int dz\,p(z)b_g\delta(\chi(z)\hat{\mathbf n},z)
=\int d\chi\,H(z)p(z)b_g\delta(\chi\hat{\mathbf n},z),
$$
so its radial kernel is $q_g=Hbp$. The <cosmological Poisson equation> in <Fourier transform> variables gives
$$
-k^2\psi(\mathbf k,a)=4\pi Ga^2\bar\rho_m\delta(\mathbf k,a)
=\frac32H_0^2\Omega_m a^{-1}\delta(\mathbf k,a).
$$
Here $\Omega_m$ denotes today's <matter density parameter>, not its time-dependent value. For a growing <linear cosmological density perturbation>, write $\delta(\mathbf k,a)=D(a)\delta_{\rm ini}(\mathbf k)$, where $D$ is the <linear growth factor>, and define the <logarithmic linear growth rate> $f=d\ln D/d\ln a$. At fixed <comoving coordinate>,
$$
\left(\frac\delta a\right)'=\mathcal H(f-1)\frac\delta a=H(f-1)\delta.
$$
Consequently
$$
\boxed{\psi'(\mathbf k,z)=\frac{3H_0^2\Omega_m}{2k^2}H(z)[1-f(z)]\delta(\mathbf k,z).}
$$
In particular, $f=1$ during <matter domination> makes the linear potential constant and eliminates this <Integrated Sachs-Wolfe effect>.
To use the supplied <Limber approximation> without hiding its scale dependence in a radial kernel, take $U=\delta$, $V=\psi'$, $q_g=Hbp$, and $q_T=2$. The corresponding three-dimensional <cross-power spectrum> is
$$
P_{\delta\psi'}(k,z)=\frac{3H_0^2\Omega_m}{2k^2}H(z)[1-f(z)]P(k,z),
$$
where $P$ is the <matter power spectrum>. Substitution gives
$$
C_\ell^{gT}=3H_0^2\Omega_m\int\frac{d\chi}{\chi^2}\,\frac{H^2(z)[1-f(z)]b_gp(z)}{k^2}P(k,z)\bigg|_{k=(\ell+1/2)/\chi}.
$$
The <Limber approximation> relation $k\chi=\ell+1/2$ cancels $\chi^{-2}k^{-2}$, yielding
$$
\boxed{C_\ell^{\delta_g\,\delta T/T}=\frac{3H_0^2\Omega_m}{(\ell+1/2)^2}\int d\chi\,H^2(z)[1-f(z)]b_gp(z)P\left(\frac{\ell+1/2}{\chi},z\right),\qquad z=z(\chi).}
$$
Both powers of $H$ are necessary: one converts the redshift selection to a distance kernel, and the other differentiates the potential in <conformal time>. The factor two in the <Integrated Sachs-Wolfe effect> cancels the factor $1/2$ from the <cosmological Poisson equation>. For positive <galaxy bias> and $f<1$, the predicted correlation is positive: an overdense region whose negative potential becomes shallower gives a positive integrated <temperature> shift. The result assumes linear, scale-independent growth and the Newtonian <cosmological Poisson equation>; the <Limber approximation> becomes less accurate at the very lowest angular multipoles.
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