The neutron-evaporation residue cross sections ${\ensuremath{\sigma}}_{2n}$, ${\ensuremath{\sigma}}_{3n}$, and ${\ensuremath{\sigma}}_{4n}$ in the hot fusion reaction ${}^{48}\mathrm{Ca}{+}^{249}\mathrm{Bk}{\ensuremath{\rightarrow}}^{297}\phantom{\rule{-0.16em}{0ex}}\phantom{\rule{-0.16em}{0ex}}{117}^{*}$ are calculated at the compound-nucleus (CN) excitation energies ${E}_{\text{CN}}^{*}=35.3$ and 39.3 MeV, by taking the superheavy-region proton magic $Z=126$ and neutron magic $N=184$, within the dynamical cluster-decay model. However, the above-stated production cross sections could not be fitted if the incoming nuclei and/or decay fragments were considered to be spherical. On the other hand, with inclusion of deformation effects up to quadrupole deformations ${\ensuremath{\beta}}_{2i}$ and with the ``optimum'' orientations ${\ensuremath{\theta}}_{i}^{\text{opt}}$, the ${\ensuremath{\sigma}}_{2n}$ and ${\ensuremath{\sigma}}_{3n}$ cross sections were fitted nicely, but ${\ensuremath{\sigma}}_{4n}$ were still underestimated by about 25$%$, indicating the possibility of some other competing evaporation residue, like ${}^{4}$He decay, contributing to 4$n$-decay cross sections. Also, a possible contribution of a fusion-fission component is predicted which, together with ${}^{4}$He decay, needs experimental verification. Other important results are that (i) the angular momentum does not show any significant influence on the fragmentation path of the $Z=117$ nucleus formed in this reaction, and (ii) the barrier modification is found to be more important for the lower-angular-momentum states, and for the lightest neutron cluster, independent of the CN excitation energy.