This is a picture of the park rental stollers.
Although I have seen kids sleeping in them, I think one of them might not be too comfortable with your child's special needs. They are low to the ground, which might make it easier or harder for her to climb in and out.
With either a GAC (Guest Assistance Card) or sticker, you would be able to bring either a park rental stroller or another type of stroller into the lines. It would be handled just the same as a wheelchair. Most attractions don't have special wheelchair entrances, but in places where there is a different entrance or different way to board with a wheelchair, you would use those.
Also, to think about, if you rent one from the park, you will/may need to figure out how to transport a tired child back to your resort. There are some places listed in the disABILITIES FAQs thread that rent strollers and special needs strollers. You could also check with her doctors/OT/PT if there is a lending closet in your area. Some places do have lending closets and would be able to lend a suitable wheelchair or stroller at no cost.
I don't know that there is a rule for people using their personal power wheelchairs while holding a child, but all I can think is that it would be rather uncomfortable for both the person doing the driving and the child. It could be dangerous because a small bump on a joystick can make a very large movement in the wheelchair, so I would not advise it. If CMs see someone driving a wheelchair or
ECV in an unsafe manner, they would do something about it.