This study is inspired by a scenario, in which the Standard Model enhanced by an additional dark-matter scalar could be extended up to the Planck scale, while accommodating the low measured value of the Higgs mass. To that end, we study a toy model for a gauge-singlet dark-matter scalar coupled to the Higgs-top-quark sector of the Standard Model. Using functional methods to derive renormalization group flow equations in that model, we examine several choices for the ultraviolet, bare potential in the Higgs-dark-matter sector. Our results indicate that the dark-matter scalar can decrease the lower bound on the Higgs mass in the Standard Model. We then use the fact that higher-order couplings which are driven to tiny values by the renormalization group flow towards low energies can easily be of order one at the ultraviolet cutoff scale. Our study indicates that the inclusion of these couplings can significantly increase the ultraviolet cutoff scale and therefore the range of validity of the model while yielding a low value for the Higgs mass in the infrared. This is achieved within a setting where the dark-matter scalar accounts for the complete dark-matter relic density in our Universe.