The renormalized coupling constants g_2k that enter the equation of state and determine nonlinear susceptibilities of the system have universal values g_2k ^* at the Curie point. We use the pseudo-ε-expansion approach to calculate them together with the ratios R_2k = g_2k/g₄ ^k-1 for the three-dimensional scalar λϕ⁴ field theory. We derive pseudo-ε-expansions for g₆ ^*, g₈ ^*, R₆ ^*, and R₈ ^* in the five-loop approximation and present numerical estimates for R₆ ^* and R₈ ^*. The higher-order coefficients of the pseudo-ε-expansions for g₆ ^* and R₆ ^* are so small that simple Padé approximants turn out to suffice for very good numerical results. Using them gives R₆ ^* = 1.650, while the recent lattice calculation gave R₆ ^* = 1.649(2). The pseudo-ε-expansions of g₈ ^* and R₈ ^* are less favorable from the numerical standpoint. Nevertheless, Padé–Borel summation of the series for R₈ ^* gives the estimate R₈ ^* = 0.890, differing only slightly from the values R₈ ^* = 0.871 and R₈ ^* = 0.857 extracted from the results of lattice and field theory calculations.