We revisit the problem of approximating minimizers of certain convex functionals subject to a convexity constraint by solutions of fourth order equations of Abreu type. This approximation problem was studied in previous works of Carlier-Radice (Approximation of variational problems with a convexity constraint by PDEs of Abreu type... Calc. Var. Partial Differential Equations. 58 (2019), no. 5, Art. 170) and the author (Singular Abreu equations and minimizers of convex functionals with a convexity constraint, arXiv:1811.02355v3, Comm. Pure Appl. Math., to appear), under the uniform convexity of both the Lagrangian and constraint barrier. By introducing a new approximating scheme, we completely remove the uniform convexity of both the Lagrangian and constraint barrier. Our analysis is applicable to variational problems motivated by the original 2D Rochet-Chon\'e model in the monopolist's problem in Economics, and variational problems arising in the analysis of wrinkling patterns in floating elastic shells in Elasticity. (read more)