Beatty says the shutdown violates Cooper's May injunction, which barred any closure until the board approved one consistent with his ruling and the court modified that order.
A lawyer for Beatty was turned away Wednesday and told the building was closed to the public and interior construction had begun for safety reasons.
The closure followed a board vote Tuesday to shut the center for renovations, while Trump said reconstruction would wait until an appeals court or the Supreme Court lets his name be included officially.
The fight extends a broader naming dispute: Cooper ruled Tuesday that the board cannot restore Trump's name to the building or create Trump memorials there without Congress.