I freeze my MMF covered cakes all the time. I just wrap well with Glad seal and wrap then put them in a cake container (I do this for a 2-3 weeks usually) and the taste is no different. When I take it out, I let it thaw in the wrappings, then take the wrapping off after a few hours and then condensation comes off on the wrapping. It may be a bit tacky to the touch, so just let it air dry for a couple of hours. Then you can put on any additional decorations (RI) or ''paint'.
I have also put MMF in the fridge, but the fridge dries cake out faster. I'd rather leave a couple days on the counter (if no perishable filling/icing) than in the fridge. But freezing (even for a couple of days) is your best bet. You extend the life of the cake! HTH