Find out how long after the stated reservation pick up time that they will hold your car/rate... I believe with Avis/Budget this policy is quite generous.
Maybe you can say you are picking it up at 23:00 on Saturday night but if they will hold your car/rate for 12 hours then pick it up at 11:00 on Sunday morning?
If you do Avis Preferred your paperwork will be prepared and ready and would still possibly show the "old" pick up time.
Whether or not you feel any of this would be ethical is your call.
However it is very common with rental cars (hotels too) to tinker rates and sometimes charge more for less. This is not an unusual practice at all. Computer pricing programs gone crazy.