You can reproject design data to a new geographic coordinate system by attaching it as a reference to a new file and performing a Fence Copy or Merge Into Master from the References dialog.

The new file should have the new GSC applied. The file with your design data should be attached using the Attachment Method of Geographic - Reprojected.

When the design data is fence copied or merged, it will be reprojected to the coordinate system of the open file.