Yes, the net force is zero. But that doesn't mean you are doing no work "against gravity".
That the system is in stable equilibrium, does indicate that the kinetic energy is constant. You are changing the configuration of the system- its state of position - against a field where the force is given by the gradient of potential energy ie. $$ \mathbf{F} = -\nabla U$$. So, you have to do work against the field concerned so as to change the state. If the system had changed its velocity, that only meant you were doing extra work for executing the change in kinetic energy.