The alternative renormalizable minimal (SO\left(10\right)) model is composed of the Yukawa couplings with (\text{10}\oplus \text{120}) Higgs fields, whereas the right-handed (RH) neutrino Majorana masses are generated via the Witten mechanism. The gauge coupling unification is achieved together with a unique pattern of the fermion masses and mixing at the grand unification scale due to additional contributions of vector-like quarks to the standard model renormalization group equations. We perform the fitting of the model to the experimental data of charged fermion masses and the CKM matrix. The best fit point is obtained with large pulls for ({m}{c}), ({m}{s}), ({m}{b}), and ({m}{\tau }). For the modifications to the minimal model by adding either ({\text{10}}^{\prime }) or ({\text{120}}^{\prime }), a large deviation for the tau mass rules out all these models. In the case with the bottom and vector-like quark mixing, the mass matrices are well fitted the charged fermions but the bound on the light neutrino mass scale excludes this scenario. To ameliorate this deficit, we consider the two-step symmetry breaking scenario, (SO\left(10\right)\to SU\left(5\right)\to SU{\left(3\right)}{C}×SU{\left(2\right)}{L}×U{\left(1\right)}{Y}), with the (SO\left(10\right)) breaking at the Planck scale leading to the radiatively generated RH neutrino Majorana masses being at the ordinary seesaw scale. For this case, we find the best fit point with ({\chi }^{2}=7.8) consistent with experimental results including the neutrino sector. The largest deviation is 2.3σ corresponding to the strange quark mass. Hence, a more precise determination of the strange quark mass can test this model. For the best fit point, we find the effective Majorana neutrino mass of ({m}{\beta \beta }=0.22) meV and the sum of light neutrino masses as (\Sigma =0.078) eV, which are consistent with the current constraints from the search for the neutrinoless double beta decay and the CMB anisotropy measurement.