Follow the instructions in my answer at Failed status and optimal value NAN in convex problem to install CVXQUAD and its exponential.m replacement for CVX’s version. I think that if you use the det_rootn version of your program, then no further modifications will be necessary to your problem in order for CVXQUAD’s Pade approximation to be invoked instead of CVX’s less reliable successive approximation method.
That may or may not resolve the solution difficulties. Let us know how it works out.
Full disclosure: I haven’t really looked at your code, so no guarantees on anything or that there aren’t serious problems somewhere other than solution difficulties due to the unreliability of CVX’s successive approximation method.