Old hoyer lifts are a good source for free or cheap heavy duty casters. And they'll roll over most anything you come across.
Maybe even use a hoyer base as your starting point...unbolt the upright pieces and your left with a very stable rolling base that you could bolt a piece of plywood or...
Congratulations to you and your daughter.
Would something like a rock climbing harness or tree climber saddle provide enough support? You could use a hoyer to attach it to and be mobile.
If you need something welded just post a notice on CL. There will be plenty of people wanting to help...