Your story sounds remarkably similar to my last flight on Shuttle LAX-LAS. I arrived early from the Far East, tried to get an earlier flight out of terminal one from 6:30 P.M. onwards, finally conceded defeat and took the 10:20 p.m. UA flight that I had been booked on all along. Trouble was it didn't rotate until midnight. Reached Las Vegas at 1 a.m., no discount coupons.
On the return, went to the airport and rented a car from Hertz. Took all of 1 minute. Drove straight through to LAX in under 4 1/2 hours, arriving 2 1/2 hours before scheduled departure. Cost? Exactly the same as a shuttle flight and my nerves stayed intact.