I hate to throw a wet blanket on what could be a really fun experience, but quite honestly, that would be a really difficult trip for a brand new pilot. I'm always congratulating newbies on other forums for obtaining their license to learn, and I really mean it. So far, the longest cross countries you've done on your own are a whole lot shorter than flying for days on end, taking into account changing weather, mechanical needs and difficulties, etc. You have so much more to learn. I'd start with experiencing a really long one or two day cross country, and seeing how that worked out.
As for whether there is any way to actually do it, renting under the circumstances you're envisioning, I can't say. You're more likely to be able to convince the school you're attending than a strange FBO, so that's the place to start. But most places who will rent for lengthy periods of time do it in a much different way--it's the number of hours at normal rates (with maybe a discount for prepayment) with a minimum number of hours per day, on a dry rate (you buy the fuel and oil). The longest I've ever rented any airplane was for 3 weeks years ago, a Turbo Arrow, but I've rented several airplanes for a few days at a time, from 3 different FBOs, all of whom used the same process. When I've rented at other FBOs, the total time involved was less, but when I asked about long term rentals, it was the same as I'd experienced.
"I have slipped the surly bonds of earth..., put out my hand and touched the face of God." J.G. Magee