Mosek 9.0 (no longer beta) does indeed work with CVX 2.1 (but CVX does not utilize its native exponential cone support). So you should upgrade from Mosek 7.0 to Mosek 9.0 and try that, with CVXQUAD. Reformulate instances of log per 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
Also see my suggestion above about experimenting with the Pade Approximation parameters, m and k.