The refrigerator is a drier environment than your kitchen so it won't add to the problem. And you'll cover the pie with plastic wrap so it doesn't absorb stray refrigerator aromas and that will keep any condensation from forming while it comes back to room temperature for service. But, yes, the crust will soften somewhat regardless.

Won't the crust get soggy if I refrigerate? Yes, it does have eggs and heavy cream.

Yes, custard pies are one of the exceptions to the it's-safe-to-leave-baked-goods-unrefrigerated rule. Protein + moisture = Danger

It's a custard pie, so you should refrigerate it to be safe. Then just take it out a couple hours before serving tomorrow to come back down to room temperature.

I'd put it in the fridge -- it's got eggs and dairy in it, I'm guessing?