Honestly, for a five week roadtrip buy a car and re-sell it at the end. Before you say no...consider that there once was, and may still be, companies that specialize in this for Aussie and Kiwi visitors on tour (their long 1-2 year travels before settling down). Mostly it was Vanagon camper vans but there was even a company out of NY that did it with just about any car.

It worked like this:

You tell them what you want in a car and put down say a $1000 deposit
They locate and buy the car
They pay basically a paralegal or process server to pick you up at the airport and sign over the car
At the end of your stay, the same paralegal'ish person meets you again, looks over the car and you sell it back to the company

On Vanagon Westphalia's it was usually about $150-200 a week in depreciation for the first four weeks or so and then it would drop to about $100 week if you mileage was not insane. For a regular recent off lease sedan I think it was about half that but the company had a flat $300 paperwork fee on top.