= Solution
At fixed $\eta$, the <recombination temperature> is determined by atomic physics and the Saha equation, not by today's CMB temperature. Under the stated approximation,
$$
\boxed{\frac{T_{{\rm dec},1\rm K}}{T_{{\rm dec},2.73\rm K}}=1}.
$$
Since $1+z_{\rm dec}=T_{\rm dec}/T_0$,
$$
\boxed{\frac{1+z_{{\rm dec},1\rm K}}
{1+z_{{\rm dec},2.73\rm K}}=2.73},
$$
and the same ratio holds for the large redshifts themselves to excellent accuracy. The comoving distance $\chi_{\rm dec}=\int_0^{z_{\rm dec}}dz/H(z)$ can nevertheless be held fixed by adjusting the late-time expansion history, for example $H_0$, $\Omega_{m,0}$, spatial curvature, or dark-energy density and equation-of-state parameters.
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