It is an amazingly beautiful area and I don't think you would have any clearance issues. I have done the entire canyon route three times, however always on motorcycle trips cross country. It is a popular rafting and canoeing river so try to travel on a weekday, especially that time of year. There is a forest service campground on the western end that is overlooking a lake. I have camped there but don't remember the name. Have a great trip.
"Certainty on any matter is not one of the human attitudes the Gods admire or tolerate."