UHaul has a very nice car hauler. Built in brakes, good maintenance, reasonable price. I have a lowered 67Shelby that drives on with no issues at all.
When I raise the suspension electronically on my family SUV, it slightly tilts the tail of the car hauler down an inch. The same effect can be found by parking the rear tires of the tow vehicle on a speed bump or low curb.
Tire straps on UHaul will suffice, cinch them down. There is also a safety chain at the rear of the hauler that loops over the rear suspension--this should not be tight since it will cut a brake or fuel line.
Be aware that the car will still bounce a little. THe hauler is designed to absorb this. After a while, you will forget that it is back there.
Happy trails.
Paul