I'm going to ask someone to allow me to use hotel elevator without key access
Why don't you have a hotel key?
Actually I'm in a hurry, otherwise I wouldn't even ask.
You should not use the elevator if you do not have key access to it. That is essentially tresspassing.
The other option is to take the stairs, it's only 3 floors.
But if you aren't allowed to use the elevator in the first place then you should definitely take the stairs.