Ive done the drive LAX to PSP and as Golfmad said, it took just over 2 hours. I wouldn't want to do the drive from PHX off of BA289 as you wont have your hire car til about 7pm and its a 3 hour plus drive.
If you could get the earlier AA flight then this might be an option.
PSP is a lovely little airport, not many better in the US and the hire car desks are in the airport.