I just wanted to mention that at 13 months, your toddler's head still very much is large for the muscles in her neck. (This is one reason bicycle carriers are not recommended under a year of age, and the reason rear facing is safer).
With a helmet on, her head will be even heavier, even with the lighter weight DOT helmets (which of course are essential). Going over bumps could cause injury, and a slip or an accident could result in spinal cord damage - this is why it's important to keep the straps tight in a carseat; young children's necks have a very hard time with whiplash. I don't think a mai tai is going to support her neck.
And not to add smarmy advice or anything, if this is really the route to the hospital - in case of, say, head injury or back injury or even vomiting - it's really not in my opinion doable. I think getting the brakes fixed on the van needs to be a priority (easy to say from the outside, I know).