It sounds like you are saying that CVX reported Status failed, and all but one constraint is satisfied. Well, if one constraint is not satisfied (within tolerance), then the solution is not feasible. But remember that CVX said the optimization failed. So what exactly are you asking?

Successive approximation method to be employed.
For improved efficiency, SDPT3 is solving the dual problem.
SDPT3 will be called several times to refine the solution.
Original size: 97 variables, 40 equality constraints
4 exponentials add 32 variables, 20 equality constraints

This is the result of the operation, that is, the value of the optimization variable can still be obtained after the failure, and it is satisfied to substitute them into the constraint.

Even if the result is feasible, it may not be optimal, or even close to optimal.

have you tried using CVXQUAD, as described in one of your previous threads? I haven’t look too closely at your program, but if your only exponential cone function is log, replace log(expression)
with -rel_entr(1,expression)
in order for CVXQUAD (if installed with its exponential.m replacement) to be invoked with Pade approximant instead of CVX’s successive approximation method. And if you have Mosek, specify that as the solver ti improve the chances of successful solution.