= Solution
An overlap estimate must use nearest magnetic-ion separations. For the <face-centered cubic lattice> of Mn, $d_{\rm Mn}=a_{\rm Mn}/\sqrt2=3.147\ \mathrm{\mathring A}$; for the <body-centered cubic lattice> of Fe, $d_{\rm Fe}=\sqrt3a_{\rm Fe}/2=2.477\ \mathrm{\mathring A}$. The supplied radial functions have inverse decay lengths $\kappa_Z=Z/(3a_B)$. Their intersite overlap therefore contains approximately $e^{-\kappa_Zd}$, up to polynomial and angular factors. The decisive exponent difference is
$$
\kappa_{\rm Mn}d_{\rm Mn}-\kappa_{\rm Fe}d_{\rm Fe}
=\frac{25(4.45/\sqrt2)-26(\sqrt3\,2.86/2)}{3(0.529)}
\simeq8.99.
$$
The <exponential orbital-tail estimate of exchange> thus gives a hopping suppression approximately $e^{-8.99}\simeq1.25\times10^{-4}$ before polynomial corrections. Keeping the supplied $Z^{7/2}r^2$ radial prefactor in a midpoint overlap proxy gives
$$
\frac{\phi_{\rm Mn}(d_{\rm Mn}/2)^2}{\phi_{\rm Fe}(d_{\rm Fe}/2)^2}
\sim\left(\frac{25}{26}\right)^7
\left(\frac{d_{\rm Mn}}{d_{\rm Fe}}\right)^4e^{-8.99}
\simeq2.5\times10^{-4}.
$$
This proxy is not an evaluated <hopping integral>, but confirms that the polynomial factors do not undo the exponential loss. If comparable Coulomb denominators and nonexponential prefactors are assumed, a kinetic-exchange scale proportional to $t^2/U$ is suppressed by roughly $10^{-8}$–$10^{-7}$. Direct intersite Coulomb exchange likewise contains products of the small tails. Coordination-number differences cannot compensate for many orders of magnitude.
By contrast, the measured ordering-temperature ratio supplied in the question is \b[$122/1043\simeq0.117$], not an exponentially tiny number. \b[Within this overlap model, direct Mn–Mn exchange cannot explain MnO's substantial magnetic ordering scale by extrapolating the Fe mechanism.] Another exchange path is required.
This is an order-of-magnitude argument, not an exact relation between an itinerant <Curie temperature> and a localized-spin <Néel temperature>. The hydrogenic functions use the printed bare $Z$ values; actual solid-state orbitals are screened and hybridized. Their literal absolute exchange scales should not be treated as quantitative predictions for Fe or MnO.
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