Welcome to the Swamp, Internet Explorer! As you've found, you cannot ExportTo the same database that the style resides in. From the help for ExportTo(): System.InvalidOperationException Thrown when the current style is in the same database as the destination database.
The Style objects have the CopyAsSibling() method, but I don't believe that will work for copying a labelstyle and it's children...you may also want to include the children's children.... What I would probably do is create a side database, use ExportTo(theSideDb), rename the styles there, then ExportTo(currentDb) from that side database back to the current dwg. Then discard the side db.
Actually, I have not tried this, but perhaps the CopyAsSibling() method will copy the style AND it's children.