How does an EU representative engage with EU DPAs on behalf of the business?

This really depends on the circumstances and they will do what is required when requested by the relevant supervisory authority or Data Protection Authority. The primary function of representatives is to be a point of contact and connection point between non EU based data controllers and processors and local supervisory authorities, but ultimately the representative will be acting on behalf of the controller or processor and will have to obviously liaise with the controller and processor and get its responses to requests that are being made. So, it is like a constant flow of communication and that is the primary role of representative is to be that source of communication.