Sorry, but I got thru about 30 seconds of that video and had to turn it off because the woman said at least 3 incorrect things, all while pointing at Wilton products that wouldn't spoil in a nuclear holocaust.
Meringue buttercreams, as well as American buttercream and cream cheese icings, are shelf stable. They absolutely can be left on the counter at room temperature. Meringue buttercreams for several days. American buttercream, depending on the recipe, can be literally *forever* and it won't rot (Crisco + powdered sugar). They will not spoil, they will not turn into Cherynoble, and you will not get anyone sick "because they have dairy in them". That's just flat wrong. The cake will mold before your buttercream will. And I've actually never personally seen a cake rot or mold, and believe me I've had cake sit on my counter for days.
However, it makes transport much easier if your cake is cold. Not only does it make it more stable because your tiers won't shift or slide around because the buttercream gets rock hard, but it also makes it much harder to dent or mar when you are loading, unloading etc. The only thing you have to be careful of is meringue buttercreams crack much easier then American buttercreams, especially if your cake shifts in transport or your cake board isn't stable enough, and in all the cakes I've transported here in San Francisco (home of really bad drivers, millions of tourists getting lost, pedestrians with death wishes, cyclists that try and make you crash, pot holes the size of Volkswaggons and massive hills), I've only had one cake slightly crack at the base. The venue was over an hour away at a winery on top of a mountain. I think the cake was fine until I got to the driveway, which was an 8% downgrade on dirt WITH freakin manmade dirt speedbumps. But I was able to completely fix it on site.
I box all my cakes, then wrap in a garbage bag to prevent them getting that refrigerator funk that real butter picks up so easily. But your home fridge might not fit a 16" or 18" box. If not, don't worry about it, just make sure you take everything else out of it (especially any fruits or veggies) and put it all in an ice chest or something, and make sure you clean it well first.
And you CAN, and most of us do, refrigerate fondant cakes.
Good luck to you, I also made my own wedding cake and was SO glad I did! Not only did it spawn a new career but people to this day still rave about how my wedding cake was the best cake they've ever had at a wedding.