Mosek is probably more reliable in general than SDPT3, and better able to deal with numerically difficult problems. But based on addition of constraints causing the problem to be reported feasible when it was reported infeasible without the added constraints, it appears that your problem may be numerically unstable.
Specifying the very large upper bound of 10000 could perhaps be causing numerical difficulties in the solver. Can you look at changing the problem formulation, and specifically, perhaps the scaling so that any upper bound for S is smaller in magnitude?
You haven’t shown us what fX) and g(X) are, including whatever numerical “sins” they may exhibit, so your problem is not reproducible. If you at least show the solver output from SDPT3 and Mosek, perhaps someone, such as one of the Mosek guys who read the forum, could better assess the situation.
What are the results of
eig(f(X)+s*g(X)) using the values of s and X produced by running CVX with the reported feasible result? If
0 <= s <= 10000 and the maximum eigenvalue is either non-positive, or a small enough positive number to satisfy you as being feasible for practical purposes (i.e., within accepted tolerance), and min eigenvalue of X is nonnegative or negative of sufficiently small magnitude, then you will have the proof in the pudding of feasibility.