The ratios of the B$_\mathrm{c}$(2S)$^+$ to B$_\mathrm{c}^+$, B$_\mathrm{c}^*$(2S)$^+$ to B$_\mathrm{c}^+$, and B$_\mathrm{c}^*$(2S)$^+$ to B$_\mathrm{c}$(2S)$^+$ production cross sections are measured in proton-proton collisions at $\sqrt{s} =$ 13 TeV, using a data sample collected by the CMS experiment at the LHC, corresponding to an integrated luminosity of 143 fb$^{-1}$. The three measurements are made in the B$_\mathrm{c}^+$ meson phase space region defined by the transverse momentum $p_\mathrm{T}$ $\gt$ 15 GeV and absolute rapidity $|y|$ $\lt$ 2.4, with the excited B$_\mathrm{c}^{(*)}$(2S)$^+$ states reconstructed through the B$_\mathrm{c}^{(*)+} \pi^+\pi^-$, followed by the B$_\mathrm{c}^+$ $\to$ J/$\psi \pi^+$ and J/$\psi$ $\to$ $\mu^+\mu^-$ decays. The B$_\mathrm{c}$(2S)$^+$ to B$_\mathrm{c}^+$, B$_\mathrm{c}^*$(2S)$^+$ to B$_\mathrm{c}^+$, and B$_\mathrm{c}^*$(2S)$^+$ to B$_\mathrm{c}$(2S)$^+$ cross section ratios, including the unknown B$_\mathrm{c}^{(*)}$(2S)$^+$ $\to$ B$_\mathrm{c}^{(*)+}\pi^+\pi^-$ branching fractions, are (3.47 $\pm$ 0.63 (stat) $\pm$ 0.33 (syst))%, (4.69 $\pm$ 0.71 (stat) $\pm$ 0.56 (syst))%, and 1.35 $\pm$ 0.32 (stat) $\pm$ 0.09 (syst), respectively. None of these ratios shows a significant dependence on the $p_\mathrm{T}$ or $|y|$ of the B$_\mathrm{c}$ meson. The normalized dipion invariant mass distributions from the decays B$_\mathrm{c}^{(*)}$(2S)$^+$ $\to$ B$_\mathrm{c}^{(*)}$$^+ \pi^+\pi^-$ are also reported.