My experience: if you ask beforehand, they will probably charge you for whatever they offer (but maybe not a full extra night).
I prefer to just show up and ask. I generally ask if I can leave my luggage. If they are not full I usually get access to a room. But I always enter the hotel knowing there is a chance the answer will be “no” (for the room, not for the luggage).

And yes, Gold or Platinum plastic might help in this case. But it is not strictly needed to get in early.