"I am wondering if there is any advantage to faxing mine shortly after midnight."
It can't hurt.
For a different national park, certain reservations opened up on one day in January at 7:00 a.m. I got my reservation in at 7:01 and was successful. Just for grins, I checked back at 7:15, and it was full.
It also helps to make a telephone call to the office where the fax machine is located, and ask them some routine question about the fax reception. That prompts the office staff to check their machine to make sure that it has paper.