Sometimes it's worth getting creative. Just because prices are high at the airport rental locations doesn't mean they are high elsewhere. I do a lot of one-way rentals between Fort Lauderdale and Orlando. Sometimes prices at the airport can be 3-5 times higher than some of the local offices. Other times, it's the reverse.
As jsrj21 noted, there is an Alamo location at the Dolphin. Alamo also has locations at the Disney Car Care Center (near MK) and at the Buena Vista Palace. The Buena Vista Palace is directly across the street from the main bus area at Disney Springs. You can also find Avis at the Hilton Lake Buena Vista, Dollar at the Wyndham Lake Buena Vista, and Budget at the DoubleTree Suites Lake Buena Vista. (All three of those are walking distance from Disney Springs.) Finally, there is a Hertz office at the Hilton Bonnet Creek.
Try pricing a one-way rental from the airport to one of the above locations. Then, get a price for getting a van there for the remainder of your trip. You may get lucky and stumble on a deal that way. The Avis, Dollar, and Alamo (at Buena Vista Palace) are all within a 5 minute walk of each other. So, you could even switch companies without too much hassle.