Luttinger-liquid parameters are extracted from ground-state energy calculations using a fermionic functional renormalization group (fRG) scheme similar to that proposed by Andergassen et al. [Phys. Rev. B 73, 045125 (2006)]. A different way of interaction renormalization is proposed and compared to the method in this reference. The proposed new interaction renormalization produces superior numerical results for chemical potential as well as compressibility, while the Luttinger parameter $K$ is obtained quantitatively similar in both interaction renormalization schemes. It is demonstrated that a consistent extraction of Luttinger-liquid parameters from the numerically calculated ground state within fRG meets several severe numerical issues, partly related to the fact that the truncated fRG is not a conserving approximation.