Element-wise is not a semidefinite constraint (unless matrix dimension = 1).
If you have a semoidefinite constraint which is not affine in the variables, then it is not an LMI, and can not be used in CVX. It might be a BMI (Bilinear Matrix Inequality, which is non convex, or a more general nonlinear SDP.
Without seeing the rest of your problem, we don’t know whether a reformulation into a problem acceptable to CVX is possible. But as now formulated, it is non-convex.
The onus is on you to prove you have a convex problem. If you have done so, people on this forum might be able to help you get entered in CVX, if that is possible. Why isn't CVX accepting my model? READ THIS FIRST! .