It should “work” if you do it correctly, which will of course require the input data (parameters) to be of the correct sign, which I assumed (otherwise the constraint will not be convex). There is not a “separate” optimization inside another. You need to add variables and constraints to your overall problem/
Given use of log)sum_exp and exp, after you verify you have the initial model accepted by CVX, you follow the advice at CVXQUAD: How to use CVXQUAD's Pade Approximant instead of CVX's unreliable Successive Approximation for GP mode, log, exp, entr, rel_entr, kl_div, log_det, det_rootn, exponential cone. CVXQUAD's Quantum (Matrix) Entropy & Matrix Log related functions
I suggest you start with a simplified problem, get that working, then attempt the problem you really want to solve. Fort instance, start with the simplified constraint in the link, then increase the complexity to the problem you really want to solve.