This is 100% OP's fault and it's only Avis' own silliness that it apologized and consented to the chargeback. OP:
1. Took, without permission, a vehicle to which he was not assigned. Whether entitled or not, the key is permission."
2. OP ought to thank the lot attendant for catching OP's scam and changing the paperwork. Else the missing vehicle would be reported stolen and OP answering some unpleasant questions to folks who carry guns.
3. OP didn't read the paperwork when he left the lot, didn't read the paperwork when he returned the vehicle and didn't read what he submitted to his own employer.
Neither Hertz nor any other car rental company will be any different. They may screw-up, but you can't use self-help to fix the problem.
Whatever it is that OP says he spent -- actually his clients spend -- this is a customer Avis can well afford to lose.