I didn't think the OP was serious about doing it herself. Note the placement of caps for EXTRA $610. She's a new member who came here only to vent about how expensive it is to get the cake of her dreams.
In my house we would avoid storage issues and finish eating the entire cake .
If it's an ice cream layer cake, it must be stored in the freezer. If it has ice cream mixed in as an ingredient and then baked, store as you would any comparable cake.