The problem with storing the cake in the fridge is once you bring it back to room temperature the fondant begins to sweat and can become moist and sticky. If the cake is cut the temperature alterations can dry the cake out and make it stodgy in the centre. If you only intended to store it for a few hours I don’t see you would have a problem as I have never experienced any for a short period.
If you are NOT using fresh cream, the cake can be kept at room temperature if it is appropriately covered and it will last up to 10 days. The cake lace as a separate item once mixed into a paste will last up to 7 days if covered and kept in the fridge, once made into lace it can be kept for several months if it is wrapped and stored properly. I store it between grease proof paper and then inside a zip lock bag.
I hope this in someway helps :) xxx
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