Welcome to the RTA Forum!
Sorry, but I just don't think I see the point of your trip. How and Why exactly do you think that driving about 1,000+ miles a day for 10 days is going to show a car company that they should give you a car to take another trip in the future?
First of all, if I was a marketing executive of a car company, you showing me that you are willing to drive 20 hours a day every day for 10 days really wouldn't impress me much. If anything, it would make me question your judgement and maturity as someone with whom I might be doing business with.
Second, if you haven't noticed, these aren't exactly booming times for the auto industry. Any sort of promotional/sponsorship is going to have to have a far more specific goal than just driving around and showcasing a fuel efficent car. You would have to have a very specific plan to show why your idea would be drawing new customers to them - and you're going to have to prove that you can do that. Proving that you can drive a large number of miles in a short amount of time really won't do anything to show that you can sell, and no company is just going to give you a sponsorship so you can drive around on a vacation on their dime.
If you've got some more specifics with your plan that perhaps I haven't thought of here, I'd love to hear about them, but otherwise, I see a lot more wishful thinking than realistic goals.