If memory serves me right, you cannot normally change Batman's costume until you have beaten the story mode at least once. You should be able to select your costume in Riddler's challenge missions, though. Once you've beaten the game once, you should be able to select your costume, in either Story Mode or Story Mode+ .
I am not aware of any cheats to change costumes on your first playthrough, but I have not been looking much. This GameFAQs question has one promising looking cheat, but I do not know if it's one that you've already tried or if it works on your system. (Because I don't know what system you are playing on.)
To wear one of the skins for your first playthrough select your game save then press Left, Left, Down, Down, Left, Left, Right, Up, Up, Down. Then begin your game. You can then pick from any of the skins you have downloaded for Batman.