The weakly first-order nature of the two-dimensional 5-state ferromagnetic Potts model poses challenges for numerical study. Using density-matrix and tensor-network renormalization group methods, we investigate these transitions of the $q$-state Potts model via the Klein bottle ratio $g$ on original and dual lattices. Finite-size scaling of $g$ as a function of transverse system size $L_y$ accurately locates the critical points for $q=3,4,5,6$. We further examine the transfer-matrix spectra and entanglement entropy, extracting central charges through toroidal and Klein bottle boundary conditions. For $q=5$, the extracted central charge ($c\approx 1.14811$) is close to the real part of the theoretical value $c_{5\mathrm{\text{-Potts}}}=1.1375\pm 0.0211i$ predicted by complex conformal field theories. The observed drift in the scaling exponent $b$ effectively distinguishes the continuous transition from the weakly first-order regime. Furthermore, the extrapolated divergence of $g$ confirms the first-order character of the $q=5$ Potts model, while for the $q=4$ model, both the CFT-predicted eigenvalue distribution and the logarithmic corrections are numerically verified.