A new stroller rental company has hit town. Fantasy Strollers provides carriage and spaceship styled strollers for 8 hours starting at $132.
They can fit two children up to 95 lbs each. Some carriages will have a bed option. The one below appears to be one of them.
I was sent this photo from Epcot (in a favorite quiet spot location). 
Let me know if this is something you would do for your child.
UPDATE: Disney has since put an official ban on large strollers like these. However, last time I was there I did see larger than normal wagon-type stroller which may have just met the size maximum size restrictions.
Good luck and see you in the parks!
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