Carrot Cake, side unfrosted, placed in a glass domed cake stand, will it last for 5 days without drying out?

It will depend on how good the cover is.

Most of the ones I have seen only sit in the stand ans are not air sealed, so the edges of the cake could dry a little, but that starts straight away anyway and most people will accept that.

My concern though is about your storage instructions and all I can say is we don't know what temperatures that cake is going to be sitting in for the next 5 days. I would suggest it be stored in a closet or cabinet as close to the floor as possible or in a fridge.

I find it is incredibly stupid for your customer to buy one that premature, and at the same time expecting it to be in perfect condition.

Chetak

If it's unfrosted, it will still dry out from the outside in. The crust will get hard and dry because you aren't protecting it well against the air. It's also going to grow a LOT of bacteria in this timeframe, and wouldn't be safe to eat. You should never leave food out more than 3 hours at room temperatures, and cakes usually no more than 12 hours. Bacteria thrives best at room temperature, in a moist environment with plenty of sugar to eat.

Haven't you ever had a loaf of bread from the bakery? Even if it's not cut, the longer it sits out, the harder and dryer it gets, despite being wrapped in paper.