I doubt the majors will be able to do much for you. Still, it is worth talking to managers at each location to see if they can do something for you (national call center agents and front-line sales representatives are not the right people to talk to; call and ask for the station/branch manager).

You'll likely have better luck with the local, non-chain companies. Unfortunately, I only know the name of one company (Beverly Hills Car Rental, one of whose managers I used to do business with), but their rates aren't exactly cheap (I think their cheapest car is $50/day, and they go up to exotic cars for well over $1,000/day). Still, local agencies have more flexibility and autonomy, and they are likely more interested in individual renters than the big chains.