Storage of cakes with edible toppers largely depends on various factors like the material of the edible image, the cake icing, and humidity. Those who put photos on cakes have contrasting experiences between them when it comes to storage. Some have said that they used a certain kind of cake icing, applied the store-brought edible image, refrigerated the cake overnight, and ended up with a differently tinted image afterwards. Others have mentioned that they stored their cakes with edible toppers in refrigerators and they turned out just fine. Edible image brands differ in opinion, too; some claim that their product can be stored with the cake in the refrigerator and still be in good shape, but others do not specify this. Humidity has been stated by some bakers as a factor too but has not really been properly backed.
Cakes decorated with edible cake toppers can definitely be refrigerated, but the image may or may not be discolored afterwards. Only the photos’ color quality is affected though because they can still be safe to eat even when refrigerated.
Still, You need to do a test run yourself and rely on your experience.