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We compute the ${\mathcal O}(\alpha_s^2)$ corrections to the exclusivechannels $e^+e^-\to J/\psi+\chi_{cJ}$ ($J=0,1,2$) at $\sqrt{s}=10.58$ GeVwithin the nonrelativistic QCD (NRQCD) factorization framework. The validity ofNRQCD factorization at ${\cal O}(\alpha_s^2)$ has been confirmed for thesedouble-charmonium exclusive production processes. We analyze the impact of the$\mathcal{O}(\alpha_s^2)$ corrections on the polarized and unpolarized crosssections, as well as the $J/\psi$ angular distributions, which largely reducethe renormalization scale dependence but increase the ${\mathcal O}(\alpha_s)$NRQCD predictions to some extent for $\chi_{c0,1}$. With high numericalaccuracy, our predictions for $\sigma(J/\psi+\chi_{c1,2})$ through ${\mathcalO}(\alpha_s^2)$ are compatible with the upper limit of the \texttt{Belle}measurement. Although the theoretical prediction for $\sigma(J/\psi+\chi_{c0})$is consistent with both \texttt{Belle} and \texttt{BaBar} measurements withinuncertainties, there still exists serious tension between the predicted and themeasured profiles for the $J/\psi$ angular distribution. Regarding the brightobservation prospect of the $e^+e^-\to J/\psi+\chi_{c1,2}$ channels in\texttt{Belle 2} experiment, it is interesting to compare the futuremeasurements with our NRQCD predictions. The more accurate measurement of$e^+e^-\to J/\psi+\chi_{0}$ at \texttt{Belle 2} will also help to clarify thelong-standing puzzle of $J/\psi$ angular distribution.

${\mathcal O}(\alpha_s^2)$ corrections to $J/\psi+\chi_{c0,1,2}$ production at $B$ factories