LOWER MANHATTAN — The long battle over the controversial construction of two residential towers inside Brooklyn Bridge Park at Pier 6 may soon be resolved by a private mediator. Manhattan Supreme Court Judge Lucy Billings on Tuesday urged the Brooklyn Heights Association — which filed a lawsuit in July against the 28-story and 14-story towers— and the Brooklyn Bridge Park Corporation, the nonprofit chartered by the city to oversee the park, to resolve their dispute through mediation. She also, however, gave them the option to return to court next Wednesday if they wanted to “spend the resources to argue [in court] while in mediation.” Many locals in Brooklyn Heights and Cobble Hill have long opposed the construction of residential development in the park, but under the park’s “general project plan,” issued in 2006, it was required to be financially self-sustaining — meaning that it would generate its own revenue to cover operating and maintenance costs.