Two-dimensional spin-orbital magnets with strong exchange frustration have recently been predicted to facilitate the realization of a quantum critical point in the Gross-Neveu-SO(3) universality class. In contrast to previously known Gross-Neveu-type universality classes, this quantum critical point separates a Dirac semimetal and a long-range-ordered phase, in which the fermion spectrum is only partially gapped out. Here, we characterize the quantum critical behavior of the Gross-Neveu-SO(3) universality class by employing three complementary field-theoretical techniques beyond their leading orders. We compute the correlation-length exponent ν, the order-parameter anomalous dimension ηφ, and the fermion anomalous dimension ηψ using a three-loop ϵ expansion around the upper critical space-time dimension of four, a second-order large-N expansion (with the fermion anomalous dimension obtained even at the third order), as well as a functional renormalization group approach in the improved local potential approximation. For the physically relevant case of N=3 flavors of two-component Dirac fermions in 2+1 space-time dimensions, we obtain the estimates 1/ν=1.03(15), ηφ=0.42(7), and ηψ=0.180(10) from averaging over the results of the different techniques, with the displayed uncertainty representing the degree of consistency among the three methods.