Solve dual of a non-convex problem

Is it possible to specify a non-convex primal problem and ask CVX to solve its dual (which is always convex) and not the primal problem? There will be some duality gap but I can deal with it separately.

No, there is not. CVX requires all problems to be convex. You must find the dual yourself, and of course it must be expressible according to CVX’s disciplined convex programming rules.